%------------------------------------------------------------------------------
% File     : ITP246_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Succ 00795_053196
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0070_VEBT_Succ_00795_053196 [Des22]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.67 v8.1.0
% Syntax   : Number of formulae    : 12210 (4601 unt;1667 typ;   0 def)
%            Number of atoms       : 24061 (9104 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 14576 (2262   ~; 517   |;1132   &)
%                                         (2766 <=>;7899  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   4 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       : 1462 ( 957 fml; 505 var)
%            Number of X terms     :  678 (   0  []; 566 ite; 112 let)
%            Number of types       :  289 ( 288 usr)
%            Number of type conns  : 1529 ( 942   >; 587   *;   0   +;   0  <<)
%            Number of predicates  :  135 ( 132 usr;   2 prp; 0-4 aty)
%            Number of functors    : 1262 (1262 usr; 452 con; 0-8 aty)
%            Number of variables   : 22753 (22108   !; 645   ?;22753   :)
% 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 01:26:24.266
%------------------------------------------------------------------------------
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tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Nat__Onat_J,type,
    fun_Ex7660298398204468809at_nat: $tType ).

tff(ty_n_t__fun_It__Extended____Nat__Oenat_Mt__Int__Oint_J,type,
    fun_Ex3482447378695272101at_int: $tType ).

tff(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

tff(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    list_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    fun_real_complex: $tType ).

tff(ty_n_t__fun_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
    fun_complex_real: $tType ).

tff(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    set_list_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    product_prod_o_nat: $tType ).

tff(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    product_prod_o_int: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    fun_list_nat_o: $tType ).

tff(ty_n_t__fun_It__List__Olist_It__Int__Oint_J_M_Eo_J,type,
    fun_list_int_o: $tType ).

tff(ty_n_t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    set_VEBT_VEBT: $tType ).

tff(ty_n_t__fun_It__fun_It__Real__Oreal_M_Eo_J_M_Eo_J,type,
    fun_fun_real_o_o: $tType ).

tff(ty_n_t__fun_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    fun_VEBT_VEBT_o: $tType ).

tff(ty_n_t__fun_It__Nat__Onat_Mt__Complex__Ocomplex_J,type,
    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__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

tff(ty_n_t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    set_Code_integer: $tType ).

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

tff(ty_n_t__List__Olist_It__Extended____Nat__Oenat_J,type,
    list_Extended_enat: $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__Product____Type__Ounit_J,type,
    set_Product_unit: $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__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__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__fun_I_Eo_Mt__fun_I_Eo_M_Eo_J_J,type,
    fun_o_fun_o_o: $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__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__Int__Oint_J,type,
    fun_o_int: $tType ).

tff(ty_n_t__Product____Type__Ounit,type,
    product_unit: $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__String__Oliteral,type,
    literal: $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 (1379)
tff(sy_c_ATP_058Lamm__a____,type,
    aTP_Lamm_a: set_Extended_enat > fun_Extended_enat_o ).

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

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

tff(sy_c_ATP_058Lamm__ac____,type,
    aTP_Lamm_ac: set_list_nat > fun_list_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: complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__ag____,type,
    aTP_Lamm_ag: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ah____,type,
    aTP_Lamm_ah: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ai____,type,
    aTP_Lamm_ai: fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__aj____,type,
    aTP_Lamm_aj: fun_Co2081214437782444048eger_o ).

tff(sy_c_ATP_058Lamm__ak____,type,
    aTP_Lamm_ak: set_Extended_enat > fun_se6072594010446775118enat_o ).

tff(sy_c_ATP_058Lamm__al____,type,
    aTP_Lamm_al: set_complex > fun_se9162176452187397432plex_o ).

tff(sy_c_ATP_058Lamm__am____,type,
    aTP_Lamm_am: set_real > fun_se2161501677553633980real_o ).

tff(sy_c_ATP_058Lamm__an____,type,
    aTP_Lamm_an: set_list_nat > fun_se1865396762483378772_nat_o ).

tff(sy_c_ATP_058Lamm__ao____,type,
    aTP_Lamm_ao: set_int > fun_se867122708194164540_int_o ).

tff(sy_c_ATP_058Lamm__ap____,type,
    aTP_Lamm_ap: set_nat > fun_se382663106265794036_nat_o ).

tff(sy_c_ATP_058Lamm__aq____,type,
    aTP_Lamm_aq: fun_real_real ).

tff(sy_c_ATP_058Lamm__ar____,type,
    aTP_Lamm_ar: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__as____,type,
    aTP_Lamm_as: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__at____,type,
    aTP_Lamm_at: fun_int_int ).

tff(sy_c_ATP_058Lamm__au____,type,
    aTP_Lamm_au: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__av____,type,
    aTP_Lamm_av: rat > fun_rat_rat ).

tff(sy_c_ATP_058Lamm__aw____,type,
    aTP_Lamm_aw: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ay____,type,
    aTP_Lamm_ay: fun_complex_complex ).

tff(sy_c_ATP_058Lamm__az____,type,
    aTP_Lamm_az: fun_real_real ).

tff(sy_c_ATP_058Lamm__ba____,type,
    aTP_Lamm_ba: fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bb____,type,
    aTP_Lamm_bb: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__bc____,type,
    aTP_Lamm_bc: fun_int_int ).

tff(sy_c_ATP_058Lamm__bd____,type,
    aTP_Lamm_bd: list_complex > fun_complex_o ).

tff(sy_c_ATP_058Lamm__be____,type,
    aTP_Lamm_be: list_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__bf____,type,
    aTP_Lamm_bf: list_list_nat > fun_list_nat_o ).

tff(sy_c_ATP_058Lamm__bg____,type,
    aTP_Lamm_bg: list_VEBT_VEBT > fun_VEBT_VEBT_o ).

tff(sy_c_ATP_058Lamm__bh____,type,
    aTP_Lamm_bh: list_o > fun_o_o ).

tff(sy_c_ATP_058Lamm__bi____,type,
    aTP_Lamm_bi: list_nat > fun_nat_o ).

tff(sy_c_ATP_058Lamm__bj____,type,
    aTP_Lamm_bj: list_int > fun_int_o ).

tff(sy_c_ATP_058Lamm__bk____,type,
    aTP_Lamm_bk: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__bl____,type,
    aTP_Lamm_bl: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__bm____,type,
    aTP_Lamm_bm: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bn____,type,
    aTP_Lamm_bn: rat > fun_nat_fun_rat_rat ).

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

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

tff(sy_c_ATP_058Lamm__bq____,type,
    aTP_Lamm_bq: complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__br____,type,
    aTP_Lamm_br: rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__bs____,type,
    aTP_Lamm_bs: real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__bt____,type,
    aTP_Lamm_bt: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bu____,type,
    aTP_Lamm_bu: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__bv____,type,
    aTP_Lamm_bv: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__bw____,type,
    aTP_Lamm_bw: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__bx____,type,
    aTP_Lamm_bx: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__by____,type,
    aTP_Lamm_by: fun_nat_int > fun_int_fun_nat_int ).

tff(sy_c_ATP_058Lamm__bz____,type,
    aTP_Lamm_bz: fun_int_int > fun_int_fun_int_int ).

tff(sy_c_ATP_058Lamm__ca____,type,
    aTP_Lamm_ca: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cb____,type,
    aTP_Lamm_cb: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__cc____,type,
    aTP_Lamm_cc: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cd____,type,
    aTP_Lamm_cd: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__ce____,type,
    aTP_Lamm_ce: fun_nat_nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__cf____,type,
    aTP_Lamm_cf: fun_nat_int > fun_na7654559027617327379at_int ).

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

tff(sy_c_ATP_058Lamm__ch____,type,
    aTP_Lamm_ch: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__ci____,type,
    aTP_Lamm_ci: fun_nat_complex > fun_na1591161823130732966omplex ).

tff(sy_c_ATP_058Lamm__cj____,type,
    aTP_Lamm_cj: fun_nat_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ck____,type,
    aTP_Lamm_ck: fun_nat_rat > fun_nat_fun_rat_rat ).

tff(sy_c_ATP_058Lamm__cl____,type,
    aTP_Lamm_cl: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cm____,type,
    aTP_Lamm_cm: fun_nat_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__cn____,type,
    aTP_Lamm_cn: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__co____,type,
    aTP_Lamm_co: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__cp____,type,
    aTP_Lamm_cp: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__cq____,type,
    aTP_Lamm_cq: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__cs____,type,
    aTP_Lamm_cs: complex > fun_na1142814060743658952omplex ).

tff(sy_c_ATP_058Lamm__ct____,type,
    aTP_Lamm_ct: real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__cu____,type,
    aTP_Lamm_cu: rat > fun_nat_fun_nat_rat ).

tff(sy_c_ATP_058Lamm__cv____,type,
    aTP_Lamm_cv: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__cw____,type,
    aTP_Lamm_cw: int > fun_nat_fun_nat_int ).

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

tff(sy_c_ATP_058Lamm__cy____,type,
    aTP_Lamm_cy: fun_nat_rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__cz____,type,
    aTP_Lamm_cz: fun_nat_nat > fun_nat_nat ).

tff(sy_c_ATP_058Lamm__da____,type,
    aTP_Lamm_da: fun_nat_int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__db____,type,
    aTP_Lamm_db: complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__dc____,type,
    aTP_Lamm_dc: rat > fun_nat_rat ).

tff(sy_c_ATP_058Lamm__dd____,type,
    aTP_Lamm_dd: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__de____,type,
    aTP_Lamm_de: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__df____,type,
    aTP_Lamm_df: int > fun_nat_int ).

tff(sy_c_ATP_058Lamm__dg____,type,
    aTP_Lamm_dg: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__dh____,type,
    aTP_Lamm_dh: fun_nat_int ).

tff(sy_c_ATP_058Lamm__di____,type,
    aTP_Lamm_di: fun_int_int ).

tff(sy_c_ATP_058Lamm__dj____,type,
    aTP_Lamm_dj: fun_nat_nat > fun_fu7002938529518989751at_nat ).

tff(sy_c_ATP_058Lamm__dk____,type,
    aTP_Lamm_dk: fun_nat_int > fun_fu8464300013002972271at_int ).

tff(sy_c_ATP_058Lamm__dl____,type,
    aTP_Lamm_dl: fun_int_int > fun_fu1143798476471108391nt_int ).

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

tff(sy_c_ATP_058Lamm__dn____,type,
    aTP_Lamm_dn: fun_nat_int > fun_nat_fun_nat_int ).

tff(sy_c_ATP_058Lamm__do____,type,
    aTP_Lamm_do: fun_int_int > fun_nat_fun_int_int ).

tff(sy_c_ATP_058Lamm__dp____,type,
    aTP_Lamm_dp: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dq____,type,
    aTP_Lamm_dq: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dr____,type,
    aTP_Lamm_dr: fun_nat_complex > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ds____,type,
    aTP_Lamm_ds: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dt____,type,
    aTP_Lamm_dt: fun_nat_real ).

tff(sy_c_ATP_058Lamm__du____,type,
    aTP_Lamm_du: fun_nat_real > fun_nat_fun_nat_real ).

tff(sy_c_ATP_058Lamm__dv____,type,
    aTP_Lamm_dv: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__dw____,type,
    aTP_Lamm_dw: fun_nat_complex > fun_co3258106519541737638omplex ).

tff(sy_c_ATP_058Lamm__dx____,type,
    aTP_Lamm_dx: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__dy____,type,
    aTP_Lamm_dy: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__dz____,type,
    aTP_Lamm_dz: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ea____,type,
    aTP_Lamm_ea: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eb____,type,
    aTP_Lamm_eb: fun_nat_real ).

tff(sy_c_ATP_058Lamm__ec____,type,
    aTP_Lamm_ec: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ed____,type,
    aTP_Lamm_ed: fun_nat_real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__ee____,type,
    aTP_Lamm_ee: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ef____,type,
    aTP_Lamm_ef: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__eg____,type,
    aTP_Lamm_eg: fun_nat_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__eh____,type,
    aTP_Lamm_eh: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ei____,type,
    aTP_Lamm_ei: fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__ej____,type,
    aTP_Lamm_ej: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__ek____,type,
    aTP_Lamm_ek: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__el____,type,
    aTP_Lamm_el: int > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__em____,type,
    aTP_Lamm_em: nat > fun_co1148497455491383662plex_o ).

tff(sy_c_ATP_058Lamm__en____,type,
    aTP_Lamm_en: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__eo____,type,
    aTP_Lamm_eo: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ep____,type,
    aTP_Lamm_ep: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__eq____,type,
    aTP_Lamm_eq: fun_nat_real ).

tff(sy_c_ATP_058Lamm__er____,type,
    aTP_Lamm_er: nat > fun_complex_o ).

tff(sy_c_ATP_058Lamm__es____,type,
    aTP_Lamm_es: real > fun_fu6899955116716589935t_real ).

tff(sy_c_ATP_058Lamm__et____,type,
    aTP_Lamm_et: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__ev____,type,
    aTP_Lamm_ev: fun_nat_real > fun_nat_real ).

tff(sy_c_ATP_058Lamm__ew____,type,
    aTP_Lamm_ew: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ex____,type,
    aTP_Lamm_ex: nat > fun_nat_complex ).

tff(sy_c_ATP_058Lamm__ey____,type,
    aTP_Lamm_ey: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ez____,type,
    aTP_Lamm_ez: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fa____,type,
    aTP_Lamm_fa: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fb____,type,
    aTP_Lamm_fb: nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fc____,type,
    aTP_Lamm_fc: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__fd____,type,
    aTP_Lamm_fd: nat > fun_na3244735409550470327at_nat ).

tff(sy_c_ATP_058Lamm__fe____,type,
    aTP_Lamm_fe: fun_nat_nat > fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ff____,type,
    aTP_Lamm_ff: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__fg____,type,
    aTP_Lamm_fg: fun_nat_nat > fun_fu4413992040155564420at_nat ).

tff(sy_c_ATP_058Lamm__fh____,type,
    aTP_Lamm_fh: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fi____,type,
    aTP_Lamm_fi: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__fj____,type,
    aTP_Lamm_fj: complex > fun_na1554462459066888140plex_o ).

tff(sy_c_ATP_058Lamm__fk____,type,
    aTP_Lamm_fk: complex > fun_real_o ).

tff(sy_c_ATP_058Lamm__fl____,type,
    aTP_Lamm_fl: num > fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fm____,type,
    aTP_Lamm_fm: num > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__fn____,type,
    aTP_Lamm_fn: fun_na4138050610385705875at_nat ).

tff(sy_c_ATP_058Lamm__fo____,type,
    aTP_Lamm_fo: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fp____,type,
    aTP_Lamm_fp: fun_real_o ).

tff(sy_c_ATP_058Lamm__fq____,type,
    aTP_Lamm_fq: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__fr____,type,
    aTP_Lamm_fr: fun_real_o ).

tff(sy_c_ATP_058Lamm__fs____,type,
    aTP_Lamm_fs: fun_real_fun_real_o ).

tff(sy_c_ATP_058Lamm__ft____,type,
    aTP_Lamm_ft: num > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fu____,type,
    aTP_Lamm_fu: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fv____,type,
    aTP_Lamm_fv: set_nat > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__fw____,type,
    aTP_Lamm_fw: code_integer > fun_Co3544761417380347684eger_o ).

tff(sy_c_ATP_058Lamm__fx____,type,
    aTP_Lamm_fx: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fy____,type,
    aTP_Lamm_fy: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__fz____,type,
    aTP_Lamm_fz: fun_nat_o ).

tff(sy_c_ATP_058Lamm__ga____,type,
    aTP_Lamm_ga: fun_nat_o ).

tff(sy_c_ATP_058Lamm__gb____,type,
    aTP_Lamm_gb: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__gc____,type,
    aTP_Lamm_gc: fun_nat_fun_nat_nat ).

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

tff(sy_c_ATP_058Lamm__ge____,type,
    aTP_Lamm_ge: rat > fun_int_o ).

tff(sy_c_ATP_058Lamm__gf____,type,
    aTP_Lamm_gf: fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gg____,type,
    aTP_Lamm_gg: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__gh____,type,
    aTP_Lamm_gh: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gi____,type,
    aTP_Lamm_gi: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__gj____,type,
    aTP_Lamm_gj: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gk____,type,
    aTP_Lamm_gk: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__gl____,type,
    aTP_Lamm_gl: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__gm____,type,
    aTP_Lamm_gm: int > fun_in2204242111582549184nt_int ).

tff(sy_c_ATP_058Lamm__gn____,type,
    aTP_Lamm_gn: rat > fun_in1696471567231710611nt_int ).

tff(sy_c_ATP_058Lamm__go____,type,
    aTP_Lamm_go: code_integer > fun_Co2470855966376689427nteger ).

tff(sy_c_ATP_058Lamm__gp____,type,
    aTP_Lamm_gp: fun_Co1092983284983208416er_int ).

tff(sy_c_ATP_058Lamm__gq____,type,
    aTP_Lamm_gq: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__gr____,type,
    aTP_Lamm_gr: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__gs____,type,
    aTP_Lamm_gs: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__gt____,type,
    aTP_Lamm_gt: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__gu____,type,
    aTP_Lamm_gu: nat > fun_na7928631014926962227_nat_o ).

tff(sy_c_ATP_058Lamm__gv____,type,
    aTP_Lamm_gv: fun_na5914473681949881652_nat_o ).

tff(sy_c_ATP_058Lamm__gw____,type,
    aTP_Lamm_gw: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__gx____,type,
    aTP_Lamm_gx: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__gy____,type,
    aTP_Lamm_gy: nat > fun_na4143055713577223196at_nat ).

tff(sy_c_ATP_058Lamm__gz____,type,
    aTP_Lamm_gz: fun_na1668100129766719005at_nat ).

tff(sy_c_ATP_058Lamm__ha____,type,
    aTP_Lamm_ha: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hb____,type,
    aTP_Lamm_hb: nat > fun_li6914387844523535256_nat_o ).

tff(sy_c_ATP_058Lamm__hc____,type,
    aTP_Lamm_hc: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hd____,type,
    aTP_Lamm_hd: num > fun_num_int ).

tff(sy_c_ATP_058Lamm__he____,type,
    aTP_Lamm_he: fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__hf____,type,
    aTP_Lamm_hf: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__hg____,type,
    aTP_Lamm_hg: fun_num_option_num ).

tff(sy_c_ATP_058Lamm__hh____,type,
    aTP_Lamm_hh: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__hi____,type,
    aTP_Lamm_hi: num > fun_nat_option_num ).

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

tff(sy_c_ATP_058Lamm__hk____,type,
    aTP_Lamm_hk: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__hl____,type,
    aTP_Lamm_hl: nat > fun_num_option_num ).

tff(sy_c_ATP_058Lamm__hm____,type,
    aTP_Lamm_hm: num > fun_nat_option_num ).

tff(sy_c_ATP_058Lamm__hn____,type,
    aTP_Lamm_hn: fun_na7595719129879163854on_num ).

tff(sy_c_ATP_058Lamm__ho____,type,
    aTP_Lamm_ho: fun_Co8521658056209057742er_num ).

tff(sy_c_ATP_058Lamm__hp____,type,
    aTP_Lamm_hp: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hq____,type,
    aTP_Lamm_hq: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hr____,type,
    aTP_Lamm_hr: nat > fun_nat_fun_nat_o ).

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

tff(sy_c_ATP_058Lamm__ht____,type,
    aTP_Lamm_ht: rat > fun_Pr3238374608701176162_int_o ).

tff(sy_c_ATP_058Lamm__hu____,type,
    aTP_Lamm_hu: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hv____,type,
    aTP_Lamm_hv: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hw____,type,
    aTP_Lamm_hw: nat > fun_na7234455742697417626_nat_o ).

tff(sy_c_ATP_058Lamm__hx____,type,
    aTP_Lamm_hx: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hy____,type,
    aTP_Lamm_hy: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__hz____,type,
    aTP_Lamm_hz: fun_nat_nat ).

tff(sy_c_ATP_058Lamm__ia____,type,
    aTP_Lamm_ia: fun_real_o ).

tff(sy_c_ATP_058Lamm__ib____,type,
    aTP_Lamm_ib: fun_real_o ).

tff(sy_c_ATP_058Lamm__ic____,type,
    aTP_Lamm_ic: fun_nat_fun_nat_nat ).

tff(sy_c_ATP_058Lamm__id____,type,
    aTP_Lamm_id: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ie____,type,
    aTP_Lamm_ie: fun_real_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__if____,type,
    aTP_Lamm_if: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ig____,type,
    aTP_Lamm_ig: fun_real_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ih____,type,
    aTP_Lamm_ih: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__ii____,type,
    aTP_Lamm_ii: fun_real_real > fun_re7681423839302404990l_real ).

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

tff(sy_c_ATP_058Lamm__ik____,type,
    aTP_Lamm_ik: fun_re4238438383274954274t_real > fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__il____,type,
    aTP_Lamm_il: fun_re4238438383274954274t_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__im____,type,
    aTP_Lamm_im: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__in____,type,
    aTP_Lamm_in: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__io____,type,
    aTP_Lamm_io: fun_nat_real > fun_real_real ).

tff(sy_c_ATP_058Lamm__ip____,type,
    aTP_Lamm_ip: fun_na6724076748692969250l_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iq____,type,
    aTP_Lamm_iq: real > fun_fu5589980185390252312t_real ).

tff(sy_c_ATP_058Lamm__ir____,type,
    aTP_Lamm_ir: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__is____,type,
    aTP_Lamm_is: fun_na6724076748692969250l_real > fun_re8542384036533109323t_real ).

tff(sy_c_ATP_058Lamm__it____,type,
    aTP_Lamm_it: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__iu____,type,
    aTP_Lamm_iu: nat > fun_fu3612569900928539882l_real ).

tff(sy_c_ATP_058Lamm__iv____,type,
    aTP_Lamm_iv: fun_na6724076748692969250l_real > fun_na5225841814499866095t_real ).

tff(sy_c_ATP_058Lamm__iw____,type,
    aTP_Lamm_iw: fun_real_real ).

tff(sy_c_ATP_058Lamm__ix____,type,
    aTP_Lamm_ix: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__iy____,type,
    aTP_Lamm_iy: fun_real_real ).

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

tff(sy_c_ATP_058Lamm__ja____,type,
    aTP_Lamm_ja: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jb____,type,
    aTP_Lamm_jb: fun_nat_real > fun_fu6899955116716589935t_real ).

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

tff(sy_c_ATP_058Lamm__jd____,type,
    aTP_Lamm_jd: fun_nat_real ).

tff(sy_c_ATP_058Lamm__je____,type,
    aTP_Lamm_je: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jf____,type,
    aTP_Lamm_jf: fun_nat_real ).

tff(sy_c_ATP_058Lamm__jg____,type,
    aTP_Lamm_jg: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jh____,type,
    aTP_Lamm_jh: real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__ji____,type,
    aTP_Lamm_ji: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jj____,type,
    aTP_Lamm_jj: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jk____,type,
    aTP_Lamm_jk: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jl____,type,
    aTP_Lamm_jl: fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jm____,type,
    aTP_Lamm_jm: fun_nat_real > fun_re4238438383274954274t_real ).

tff(sy_c_ATP_058Lamm__jn____,type,
    aTP_Lamm_jn: fun_nat_real > fun_fu6119738925743952495t_real ).

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

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

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

tff(sy_c_ATP_058Lamm__jr____,type,
    aTP_Lamm_jr: fun_nat_o > fun_nat_o ).

tff(sy_c_ATP_058Lamm__js____,type,
    aTP_Lamm_js: fun_nat_o > fun_nat_fun_nat_o ).

tff(sy_c_ATP_058Lamm__jt____,type,
    aTP_Lamm_jt: fun_real_real > fun_fu3022329567491847689real_o ).

tff(sy_c_ATP_058Lamm__ju____,type,
    aTP_Lamm_ju: fun_real_real > fun_fu3077872570260440615l_real ).

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

tff(sy_c_ATP_058Lamm__jw____,type,
    aTP_Lamm_jw: fun_real_real > fun_real_o ).

tff(sy_c_ATP_058Lamm__jx____,type,
    aTP_Lamm_jx: fun_real_real > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__jy____,type,
    aTP_Lamm_jy: fun_real_real ).

tff(sy_c_ATP_058Lamm__jz____,type,
    aTP_Lamm_jz: fun_na6724076748692969250l_real ).

tff(sy_c_ATP_058Lamm__ka____,type,
    aTP_Lamm_ka: fun_re7681423839302404990l_real ).

tff(sy_c_ATP_058Lamm__kb____,type,
    aTP_Lamm_kb: nat > fun_fu3077872570260440615l_real ).

tff(sy_c_ATP_058Lamm__kc____,type,
    aTP_Lamm_kc: fun_re7681423839302404990l_real ).

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

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

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

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

tff(sy_c_ATP_058Lamm__kh____,type,
    aTP_Lamm_kh: int > fun_in5570076352459579423_int_o ).

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

tff(sy_c_ATP_058Lamm__kj____,type,
    aTP_Lamm_kj: int > fun_in5570076352459579423_int_o ).

tff(sy_c_ATP_058Lamm__kk____,type,
    aTP_Lamm_kk: rat > fun_int_fun_int_o ).

tff(sy_c_ATP_058Lamm__kl____,type,
    aTP_Lamm_kl: int > fun_int_fun_int_o ).

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    bNF_re769765379126235449at_rat: ( fun_fu6714665071556184448_rat_o * fun_fu1950632212952468284_rat_o * fun_fu6579618355643354601at_rat * fun_fu6579618355643354601at_rat ) > $o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: fun_nat_fun_nat_nat ).

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

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

tff(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
    gbinomial_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
    gbinomial_real: fun_re4238438383274954274t_real ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun_Pr9018117161326028078_int_o ).

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

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun_Pr4799464687642144302_num_o ).

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: num > fun_num_num ).

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Nat__Onat,type,
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tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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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,
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tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num,type,
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tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
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tff(sy_c_Code__Numeral_Obit__cut__integer,type,
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tff(sy_c_Code__Numeral_Odivmod__abs,type,
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tff(sy_c_Code__Numeral_Odivmod__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__num,type,
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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,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Real__Oreal,type,
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tff(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Extended____Nat__Oenat,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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tff(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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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,
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tff(sy_c_Complex_Oimaginary__unit,type,
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tff(sy_c_Complex_Orcis,type,
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tff(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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tff(sy_c_Deriv_Ohas__vector__derivative_001t__Real__Oreal,type,
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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,
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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,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Nat__Onat,type,
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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__Nat__Onat,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Nat__Onat,type,
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tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Int__Oint,type,
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tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
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tff(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment_001t__Int__Oint,type,
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tff(sy_c_Extended__Nat_OeSuc,type,
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tff(sy_c_Extended__Nat_Oenat,type,
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tff(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
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tff(sy_c_Extended__Nat_Othe__enat,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
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tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
    semiri1406184849735516958ct_int: fun_nat_int ).

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__Rat__Orat,type,
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tff(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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tff(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
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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,
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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,
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tff(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
    semiri8010041392384452111omplex: fun_nat_complex ).

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

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
    semiri1314217659103216013at_int: fun_nat_int ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
    semiri1316708129612266289at_nat: fun_nat_nat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
    semiri681578069525770553at_rat: fun_nat_rat ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
    semiri5074537144036343181t_real: fun_nat_real ).

tff(sy_c_Nat_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__Complex__Ocomplex_J,type,
    size_s3451745648224563538omplex: list_complex > nat ).

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Extended____Nat__Oenat_J,type,
    size_s3941691890525107288d_enat: list_Extended_enat > nat ).

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

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

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

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

tff(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    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__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__VEBT____Definitions__OVEBT,type,
    size_size_VEBT_VEBT: fun_VEBT_VEBT_nat ).

tff(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

tff(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: fun_nat_fun_nat_o ).

tff(sy_c_Nat__Bijection_Olist__encode,type,
    nat_list_encode: list_nat > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun_li1025265955262268170_nat_o ).

tff(sy_c_Nat__Bijection_Oprod__decode,type,
    nat_prod_decode: nat > product_prod_nat_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_Oprod__encode,type,
    nat_prod_encode: fun_Pr2602117396471556196at_nat ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: fun_nat_set_nat ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun_set_nat_nat ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: fun_nat_nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: fun_na6724076748692969250l_real ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun_real_real ).

tff(sy_c_Num_OBitM,type,
    bitM: fun_num_num ).

tff(sy_c_Num_Oinc,type,
    inc: fun_num_num ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex,type,
    neg_nu7009210354673126013omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
    neg_numeral_dbl_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Rat__Orat,type,
    neg_numeral_dbl_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl_001t__Real__Oreal,type,
    neg_numeral_dbl_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
    neg_nu6511756317524482435omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
    neg_nu3179335615603231917ec_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
    neg_nu6075765906172075777c_real: fun_real_real ).

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

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
    neg_nu8557863876264182079omplex: fun_complex_complex ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
    neg_nu5851722552734809277nc_int: fun_int_int ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
    neg_nu5219082963157363817nc_rat: fun_rat_rat ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
    neg_nu8295874005876285629c_real: fun_real_real ).

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

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun_num_num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun_num_num ).

tff(sy_c_Num_Onum_OOne,type,
    one: num ).

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

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: fun_num_nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: fun_nat_num ).

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

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

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

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
    numeral_numeral_int: fun_num_int ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: fun_num_nat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
    numeral_numeral_rat: fun_num_rat ).

tff(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
    numeral_numeral_real: fun_num_real ).

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

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: fun_num_nat ).

tff(sy_c_Num_Osqr,type,
    sqr: fun_num_num ).

tff(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

tff(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

tff(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

tff(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

tff(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: ( $o * fun_Pr4796943543331071514_nat_o * option4927543243414619207at_nat ) > $o ).

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

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

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

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

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

tff(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    bot_bo7653980558646680370d_enat: set_Extended_enat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

tff(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

tff(sy_c_Orderings_Obot__class_Obot_001t__fun_It__Nat__Onat_M_Eo_J,type,
    bot_bot_fun_nat_o: fun_nat_o ).

tff(sy_c_Orderings_Oord__class_OLeast_001t__Extended____Nat__Oenat,type,
    ord_Le1955565732374568822d_enat: fun_Extended_enat_o > extended_enat ).

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Code____Numeral__Ointeger,type,
    ord_le6747313008572928689nteger: fun_Co2081214437782444048eger_o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
    ord_less_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
    ord_less_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Rat__Orat,type,
    ord_less_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
    ord_less_real: fun_real_fun_real_o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le2529575680413868914d_enat: ( set_Extended_enat * set_Extended_enat ) > $o ).

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

tff(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: ( set_nat * set_nat ) > $o ).

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

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

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

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

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4409593196786536943plex_o: ( fun_complex_o * fun_complex_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Extended____Nat__Oenat_M_Eo_J,type,
    ord_le674264225683014473enat_o: ( fun_Extended_enat_o * fun_Extended_enat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Int__Oint_M_Eo_J,type,
    ord_less_fun_int_o: ( fun_int_o * fun_int_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Nat__Onat_M_Eo_J,type,
    ord_less_fun_nat_o: ( fun_nat_o * fun_nat_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless_001t__fun_It__Real__Oreal_M_Eo_J,type,
    ord_less_fun_real_o: ( fun_real_o * fun_real_o ) > $o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
    ord_le3102999989581377725nteger: fun_Co2081214437782444048eger_o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
    ord_less_eq_int: fun_int_fun_int_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: fun_nat_fun_nat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
    ord_less_eq_num: fun_num_fun_num_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: fun_rat_fun_rat_o ).

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
    ord_less_eq_real: fun_real_fun_real_o ).

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

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

tff(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
    ord_le7203529160286727270d_enat: ( set_Extended_enat * set_Extended_enat ) > $o ).

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

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

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

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

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

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Extended____Nat__Oenat,type,
    ord_ma741700101516333627d_enat: fun_Ex8341929632020906428d_enat ).

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

tff(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: fun_nat_fun_nat_nat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Extended____Nat__Oenat,type,
    ord_mi8085742599997312461d_enat: fun_Ex8341929632020906428d_enat ).

tff(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Orderings_Oorder__class_Oantimono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_9091379641038594480t_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    order_4130057895858720880d_enat: fun_Ex5852502975177633731d_enat > $o ).

tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Real__Oreal,type,
    order_mono_nat_real: fun_nat_real > $o ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Real__Oreal_001t__Real__Oreal,type,
    order_7092887310737990675l_real: fun_real_real > $o ).

tff(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( fun_nat_fun_nat_o * fun_nat_fun_nat_o ) > fun_nat_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Extended____Nat__Oenat,type,
    top_to3028658606643905974d_enat: extended_enat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

tff(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

tff(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
    power_power_complex: complex > fun_nat_complex ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
    power_power_nat: fun_nat_fun_nat_nat ).

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

tff(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
    power_power_real: fun_re4238438383274954274t_real ).

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

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

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

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

tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: ( int * int ) > product_prod_int_int ).

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

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

tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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tff(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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tff(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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tff(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc6161850002892822231at_nat: ( product_prod_nat_nat * product_prod_nat_nat ) > produc859450856879609959at_nat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: ( vEBT_VEBT * extended_enat ) > produc7272778201969148633d_enat ).

tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc2818902965508115497at_nat: ( fun_nat_fun_nat_nat * produc7248412053542808358at_nat ) > produc9217583074051984497at_nat ).

tff(sy_c_Product__Type_OPair_001t__fun_It__Nat__Onat_Mt__fun_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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tff(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: ( set_nat * fun_nat_set_nat ) > set_Pr1261947904930325089at_nat ).

tff(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__fun_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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tff(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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tff(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

tff(sy_c_Rat_OAbs__Rat,type,
    abs_Rat: fun_Pr1652463973008025892nt_rat ).

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

tff(sy_c_Rat_OFrct,type,
    frct: product_prod_int_int > rat ).

tff(sy_c_Rat_ORep__Rat,type,
    rep_Rat: fun_ra4503298312962071866nt_int ).

tff(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

tff(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
    field_7254667332652039916t_real: fun_rat_real ).

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

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun_Pr7537129237977221859_rat_o ).

tff(sy_c_Rat_Opositive,type,
    positive: fun_rat_o ).

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

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun_Pr9018117161326028078_int_o ).

tff(sy_c_Real_OReal,type,
    real2: fun_fun_nat_rat_real ).

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

tff(sy_c_Real_Ocr__real,type,
    cr_real: fun_fu1771228864347509049real_o ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun_fu1771228864347509049real_o ).

tff(sy_c_Real_Opositive,type,
    positive2: fun_real_o ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun_fu6714665071556184448_rat_o ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun_real_fun_nat_rat ).

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

tff(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: fun_real_real > $o ).

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

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: fun_re7681423839302404990l_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
    real_V1022390504157884413omplex: fun_complex_real ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
    real_V7735802525324610683m_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
    real_V4546457046886955230omplex: fun_real_complex ).

tff(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: fun_real_real ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
    real_V2046097035970521341omplex: real > fun_complex_complex ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
    real_V1485227260804924795R_real: fun_re7681423839302404990l_real ).

tff(sy_c_Relation_Otransp_001t__fun_It__Nat__Onat_Mt__Rat__Orat_J,type,
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tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
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tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > fun_complex_complex ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
    divide_divide_rat: rat > fun_rat_rat ).

tff(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Rat__Orat,type,
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tff(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
    dvd_dvd_real: fun_real_fun_real_o ).

tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Complex__Ocomplex,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
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tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Rat__Orat,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Real__Oreal,type,
    zero_n3304061248610475627l_real: $o > real ).

tff(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
    suminf_complex: fun_nat_complex > complex ).

tff(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: fun_nat_real > real ).

tff(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
    summable_complex: fun_nat_complex > $o ).

tff(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: fun_nat_real > $o ).

tff(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: fun_nat_real > fun_real_o ).

tff(sy_c_Set_OCollect_001_Eo,type,
    collect_o: fun_o_o > set_o ).

tff(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
    collect_Code_integer: fun_Code_integer_o > set_Code_integer ).

tff(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: fun_complex_o > set_complex ).

tff(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
    collec4429806609662206161d_enat: fun_Extended_enat_o > set_Extended_enat ).

tff(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: fun_int_o > set_int ).

tff(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: fun_list_nat_o > set_list_nat ).

tff(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: fun_nat_o > set_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: fun_Pr7716844367077540446plex_o > set_Pr5085853215250843933omplex ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    collec213857154873943460nt_int: fun_Pr3238374608701176162_int_o > set_Pr958786334691620121nt_int ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    collec3392354462482085612at_nat: fun_Pr4796943543331071514_nat_o > set_Pr1261947904930325089at_nat ).

tff(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    collec3799799289383736868l_real: fun_Pr4814023175418636514real_o > set_Pr6218003697084177305l_real ).

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

tff(sy_c_Set_OCollect_001t__VEBT____Definitions__OVEBT,type,
    collect_VEBT_VEBT: fun_VEBT_VEBT_o > set_VEBT_VEBT ).

tff(sy_c_Set_Oimage_001t__Extended____Nat__Oenat_001t__Extended____Nat__Oenat,type,
    image_80655429650038917d_enat: ( fun_Ex5852502975177633731d_enat * set_Extended_enat ) > set_Extended_enat ).

tff(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( fun_int_int * set_int ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
    image_nat_int: ( fun_nat_int * set_nat ) > set_int ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
    image_nat_nat: ( fun_nat_nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Real__Oreal,type,
    image_nat_real: ( fun_nat_real * set_nat ) > set_real ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( fun_nat_set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
    image_nat_char: ( fun_nat_char * set_nat ) > set_char ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( fun_re3844104906058862192omplex * set_real ) > set_fi4554929511873752355omplex ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( fun_re5512370375968561900l_real * set_real ) > set_fi7789364187291644575l_real ).

tff(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( fun_real_real * set_real ) > set_real ).

tff(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( fun_char_nat * set_char ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Int__Oint,type,
    insert_int: ( int * set_int ) > set_int ).

tff(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: ( nat * set_nat ) > set_nat ).

tff(sy_c_Set_Oinsert_001t__Real__Oreal,type,
    insert_real: ( real * set_real ) > set_real ).

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

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Complex__Ocomplex,type,
    set_fo1517530859248394432omplex: ( fun_na1591161823130732966omplex * nat * nat ) > fun_complex_complex ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Int__Oint,type,
    set_fo2581907887559384638at_int: ( fun_nat_fun_int_int * nat * nat ) > fun_int_int ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Nat__Onat,type,
    set_fo2584398358068434914at_nat: ( fun_nat_fun_nat_nat * nat ) > fun_nat_fun_nat_nat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Rat__Orat,type,
    set_fo1949268297981939178at_rat: ( fun_nat_fun_rat_rat * nat * nat ) > fun_rat_rat ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat_001t__Real__Oreal,type,
    set_fo3111899725591712190t_real: ( fun_na6724076748692969250l_real * nat ) > fun_na6724076748692969250l_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Extended____Nat__Oenat,type,
    set_or5403411693681687835d_enat: ( extended_enat * extended_enat ) > set_Extended_enat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
    set_or1266510415728281911st_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
    set_or1269000886237332187st_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Num__Onum,type,
    set_or7049704709247886629st_num: ( num * num ) > set_num ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Rat__Orat,type,
    set_or633870826150836451st_rat: ( rat * rat ) > set_rat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Real__Oreal,type,
    set_or1222579329274155063t_real: ( real * real ) > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: ( set_nat * set_nat ) > set_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: ( int * int ) > set_int ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Nat__Onat,type,
    set_ord_atLeast_nat: fun_nat_set_nat ).

tff(sy_c_Set__Interval_Oord__class_OatLeast_001t__Real__Oreal,type,
    set_ord_atLeast_real: real > set_real ).

tff(sy_c_Set__Interval_Oord__class_OatMost_001t__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_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: ( literal * fun_Pr1792556611952799308t_real ) > real ).

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

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

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

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
    comm_s629917340098488124ar_nat: fun_char_nat ).

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

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
    unique3096191561947761185of_nat: fun_nat_char ).

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

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

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

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

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

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

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: fun_nat_complex > $o ).

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

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun_real_real ).

tff(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
    arcosh_real: fun_real_real ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun_real_real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun_real_real ).

tff(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
    arsinh_real: fun_real_real ).

tff(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
    artanh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
    cos_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
    cos_real: fun_real_real ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Ocosh_001t__Complex__Ocomplex,type,
    cosh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
    cosh_real: fun_real_real ).

tff(sy_c_Transcendental_Ocot_001t__Complex__Ocomplex,type,
    cot_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: fun_real_real ).

tff(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
    exp_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
    exp_real: fun_real_real ).

tff(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
    ln_ln_real: fun_real_real ).

tff(sy_c_Transcendental_Olog,type,
    log: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
    powr_real: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real2: fun_re7681423839302404990l_real ).

tff(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
    sin_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
    sin_real: fun_real_real ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: fun_nat_real ).

tff(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
    sinh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
    sinh_real: fun_real_real ).

tff(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
    tan_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
    tan_real: fun_real_real ).

tff(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
    tanh_complex: fun_complex_complex ).

tff(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
    tanh_real: fun_real_real ).

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

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

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

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

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun_VEBT_VEBT_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun_Pr1263128411319240110enat_o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: fun_nat_fun_nat_nat ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun_Pr8877776288780692068_nat_o ).

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

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

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

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun_nat_fun_nat_o ).

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

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun_Pr8877776288780692068_nat_o ).

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

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: fun_VEBT_VEBT_o ).

tff(sy_c_VEBT__Member_OVEBT__internal_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: ( option_nat * option_nat ) > $o ).

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

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

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: set_nat > fun_nat_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_Opower,type,
    vEBT_VEBT_power: fun_op7210469962420818650on_nat ).

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

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun_VE2962686792283380846VEBT_o ).

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

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
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tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
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tff(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
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tff(sy_c_Wellfounded_Oaccp_001t__Nat__Onat,type,
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    accp_P1096762738010456898nt_int: ( fun_Pr9018117161326028078_int_o * product_prod_int_int ) > $o ).

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tff(sy_c_Wellfounded_Oless__than,type,
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tff(sy_c_Wellfounded_Opred__nat,type,
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tff(sy_c_aa_001_Eo_001_Eo,type,
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tff(sy_c_aa_001_Eo_001t__fun_I_Eo_M_Eo_J,type,
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tff(sy_c_aa_001t__List__Olist_I_Eo_J_001_Eo,type,
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tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001_Eo,type,
    aa_fun175815717583936595_rat_o: ( fun_fu4846507671440591850_rat_o * fun_fu8328862064278946247at_rat ) > $o ).

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

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__fun_It__Real__Oreal_Mt__Real__Oreal_J_M_Eo_J,type,
    aa_fun1584707981518017559real_o: ( fun_fu6910828598506988974real_o * fun_fu8328862064278946247at_rat ) > fun_fun_real_real_o ).

tff(sy_c_aa_001t__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_001t__fun_It__fun_It__fun_It__Nat__Onat_Mt__Rat__Orat_J_Mt__fun_It__Nat__Onat_Mt__Rat__Orat_J_J_M_Eo_J,type,
    aa_fun9037873928781546661_rat_o: ( fun_fu1950632212952468284_rat_o * fun_fu8328862064278946247at_rat ) > fun_fu4846507671440591850_rat_o ).

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

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_fequal_001t__Nat__Onat,type,
    fequal_nat: fun_nat_fun_nat_o ).

tff(sy_c_fequal_001t__Num__Onum,type,
    fequal_num: fun_num_fun_num_o ).

tff(sy_c_fequal_001t__Real__Oreal,type,
    fequal_real: fun_real_fun_real_o ).

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

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

tff(sy_c_member_001t__Complex__Ocomplex,type,
    member_complex: ( complex * set_complex ) > $o ).

tff(sy_c_member_001t__Extended____Nat__Oenat,type,
    member_Extended_enat: ( extended_enat * set_Extended_enat ) > $o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int: ( int * set_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__Nat__Onat,type,
    member_nat: ( nat * set_nat ) > $o ).

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

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

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

tff(sy_c_member_001t__Rat__Orat,type,
    member_rat: ( rat * set_rat ) > $o ).

tff(sy_c_member_001t__Real__Oreal,type,
    member_real: ( real * set_real ) > $o ).

tff(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: ( set_nat * set_set_nat ) > $o ).

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

tff(sy_v_deg____,type,
    deg: nat ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_miny____,type,
    miny: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_res____,type,
    res: nat ).

tff(sy_v_sc____,type,
    sc: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_treeList____,type,
    treeList: list_VEBT_VEBT ).

tff(sy_v_xa____,type,
    xa: nat ).

tff(sy_v_za____,type,
    za: nat ).

% Relevant facts (10531)
tff(fact_0_bbbb,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% bbbb
tff(fact_1__092_060open_0622_A_092_060le_062_Adeg_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg) ).

% \<open>2 \<le> deg\<close>
tff(fact_2_aaaa,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% aaaa
tff(fact_3_max__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_max_in_set(Xs),Xb)
    <=> ( member_nat(Xb,Xs)
        & ! [X: nat] :
            ( member_nat(X,Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_4_min__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_min_in_set(Xs),Xb)
    <=> ( member_nat(Xb,Xs)
        & ! [X: nat] :
            ( member_nat(X,Xs)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_5__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = na ).

% \<open>deg div 2 = n\<close>
tff(fact_6_semiring__norm_I85_J,axiom,
    ! [Mb: num] : aa_num_num(bit0,Mb) != one ).

% semiring_norm(85)
tff(fact_7_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit0,Nb) ).

% semiring_norm(83)
tff(fact_8_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_9_numeral__Bit0__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_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_10_high__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% high_def
tff(fact_11__092_060open_062high_Az_A_Ideg_Adiv_A2_J_A_060_Asc_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),sc) ).

% \<open>high z (deg div 2) < sc\<close>
tff(fact_12_divide__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(one)) = A ).

% divide_numeral_1
tff(fact_13_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_14_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_15_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_16_semiring__norm_I87_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit0,Mb) = aa_num_num(bit0,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(87)
tff(fact_17_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( numera6690914467698888265omplex(Mb) = numera6690914467698888265omplex(Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_18_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_real(numeral_numeral_real,Mb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Mb = Nb ) ) ).

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

% numeral_eq_iff
tff(fact_20_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_nat(numeral_numeral_nat,Mb) = aa_num_nat(numeral_numeral_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_21_numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_int(numeral_numeral_int,Mb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% numeral_eq_iff
tff(fact_22_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_23_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_24_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_25_numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% numeral_le_iff
tff(fact_26_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_27_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_28_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_29_numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% numeral_less_iff
tff(fact_30__092_060open_062z_A_060_A2_A_094_Adeg_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,za),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% \<open>z < 2 ^ deg\<close>
tff(fact_31__C5_Ohyps_C_I10_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),deg)) ).

% "5.hyps"(10)
tff(fact_32_verit__comp__simplify1_I3_J,axiom,
    ! [B: real,A2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B),A2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A2),B) ) ).

% verit_comp_simplify1(3)
tff(fact_33_verit__comp__simplify1_I3_J,axiom,
    ! [B: rat,A2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B),A2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A2),B) ) ).

% verit_comp_simplify1(3)
tff(fact_34_verit__comp__simplify1_I3_J,axiom,
    ! [B: num,A2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B),A2)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,A2),B) ) ).

% verit_comp_simplify1(3)
tff(fact_35_verit__comp__simplify1_I3_J,axiom,
    ! [B: nat,A2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A2),B) ) ).

% verit_comp_simplify1(3)
tff(fact_36_verit__comp__simplify1_I3_J,axiom,
    ! [B: int,A2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A2),B) ) ).

% verit_comp_simplify1(3)
tff(fact_37_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ord_less_eq_set_nat(A,A) ).

% verit_comp_simplify1(2)
tff(fact_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_verit__la__disequality,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
      | ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% verit_la_disequality
tff(fact_48_verit__la__disequality,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
      | ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ).

% verit_la_disequality
tff(fact_49_verit__la__disequality,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
      | ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% verit_la_disequality
tff(fact_50_verit__la__disequality,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
      | ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% verit_la_disequality
tff(fact_51_div__le__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ).

% div_le_dividend
tff(fact_52_div__le__mono,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),K)) ) ).

% div_le_mono
tff(fact_53_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] : one != aa_num_num(bit0,X2) ).

% verit_eq_simplify(10)
tff(fact_54__092_060open_062high_Ares_A_Ideg_Adiv_A2_J_A_061_Asc_092_060close_062,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,res),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = sc ).

% \<open>high res (deg div 2) = sc\<close>
tff(fact_55__092_060open_062x_A_060_Ares_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),res) ).

% \<open>x < res\<close>
tff(fact_56__092_060open_062sc_A_060_A2_A_094_Am_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,sc),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m)) ).

% \<open>sc < 2 ^ m\<close>
tff(fact_57_power2__nat__le__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_58_power2__nat__le__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_59_self__le__ge2__pow,axiom,
    ! [K: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)) ) ).

% self_le_ge2_pow
tff(fact_60_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_61_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ).

% high_bound_aux
tff(fact_62_power__minus__is__div,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),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),B2)) = 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))),B2)) ) ) ).

% power_minus_is_div
tff(fact_63_pow__sum,axiom,
    ! [A: nat,B2: 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),B2))),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))),B2) ).

% pow_sum
tff(fact_64__092_060open_0621_A_092_060le_062_An_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),na) ).

% \<open>1 \<le> n\<close>
tff(fact_65__092_060open_062mi_A_092_060le_062_Ax_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),xa) ).

% \<open>mi \<le> x\<close>
tff(fact_66__C5_Ohyps_C_I9_J,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,mi),ma) ).

% "5.hyps"(9)
tff(fact_67__C5_Ohyps_C_I6_J,axiom,
    deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,na),m) ).

% "5.hyps"(6)
tff(fact_68__092_060open_062z_A_092_060noteq_062_Ami_092_060close_062,axiom,
    za != mi ).

% \<open>z \<noteq> mi\<close>
tff(fact_69__092_060open_062mi_A_092_060noteq_062_Ama_092_060close_062,axiom,
    mi != ma ).

% \<open>mi \<noteq> ma\<close>
tff(fact_70__092_060open_062res_A_092_060le_062_Ama_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,res),ma) ).

% \<open>res \<le> ma\<close>
tff(fact_71_mem__Collect__eq,axiom,
    ! [A: extended_enat,P: fun_Extended_enat_o] :
      ( member_Extended_enat(A,collec4429806609662206161d_enat(P))
    <=> aa_Extended_enat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_72_mem__Collect__eq,axiom,
    ! [A: complex,P: fun_complex_o] :
      ( member_complex(A,collect_complex(P))
    <=> aa_complex_o(P,A) ) ).

% mem_Collect_eq
tff(fact_73_mem__Collect__eq,axiom,
    ! [A: real,P: fun_real_o] :
      ( member_real(A,collect_real(P))
    <=> aa_real_o(P,A) ) ).

% mem_Collect_eq
tff(fact_74_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_75_mem__Collect__eq,axiom,
    ! [A: nat,P: fun_nat_o] :
      ( member_nat(A,collect_nat(P))
    <=> aa_nat_o(P,A) ) ).

% mem_Collect_eq
tff(fact_76_mem__Collect__eq,axiom,
    ! [A: int,P: fun_int_o] :
      ( member_int(A,collect_int(P))
    <=> aa_int_o(P,A) ) ).

% mem_Collect_eq
tff(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_Extended_enat] : collec4429806609662206161d_enat(aTP_Lamm_a(A3)) = A3 ).

% Collect_mem_eq
tff(fact_78_Collect__mem__eq,axiom,
    ! [A3: set_complex] : collect_complex(aTP_Lamm_aa(A3)) = A3 ).

% Collect_mem_eq
tff(fact_79_Collect__mem__eq,axiom,
    ! [A3: set_real] : collect_real(aTP_Lamm_ab(A3)) = A3 ).

% Collect_mem_eq
tff(fact_80_Collect__mem__eq,axiom,
    ! [A3: set_list_nat] : collect_list_nat(aTP_Lamm_ac(A3)) = A3 ).

% Collect_mem_eq
tff(fact_81_Collect__mem__eq,axiom,
    ! [A3: set_nat] : collect_nat(aTP_Lamm_ad(A3)) = A3 ).

% Collect_mem_eq
tff(fact_82_Collect__mem__eq,axiom,
    ! [A3: set_int] : collect_int(aTP_Lamm_ae(A3)) = A3 ).

% Collect_mem_eq
tff(fact_83_Collect__cong,axiom,
    ! [P: fun_complex_o,Q: fun_complex_o] :
      ( ! [X3: complex] :
          ( aa_complex_o(P,X3)
        <=> aa_complex_o(Q,X3) )
     => ( collect_complex(P) = collect_complex(Q) ) ) ).

% Collect_cong
tff(fact_84_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_85_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_86_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_87_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_88__092_060open_062mi_A_060_Ares_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),res) ).

% \<open>mi < res\<close>
tff(fact_89_add__numeral__left,axiom,
    ! [V: num,W: num,Zb: complex] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(W)),Zb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(aa_num_num(plus_plus_num(V),W))),Zb) ).

% add_numeral_left
tff(fact_90_add__numeral__left,axiom,
    ! [V: num,W: num,Zb: 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)),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(V),W))),Zb) ).

% add_numeral_left
tff(fact_91_add__numeral__left,axiom,
    ! [V: num,W: num,Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,W)),Zb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W))),Zb) ).

% add_numeral_left
tff(fact_92_add__numeral__left,axiom,
    ! [V: num,W: num,Zb: 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)),Zb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(V),W))),Zb) ).

% add_numeral_left
tff(fact_93_add__numeral__left,axiom,
    ! [V: num,W: num,Zb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,W)),Zb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W))),Zb) ).

% add_numeral_left
tff(fact_94_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_95_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_96_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_97_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_98_numeral__plus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% numeral_plus_numeral
tff(fact_99_power__one,axiom,
    ! [Nb: nat] : aa_nat_rat(power_power_rat(one_one_rat),Nb) = one_one_rat ).

% power_one
tff(fact_100_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_101_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_102_power__one,axiom,
    ! [Nb: nat] : aa_nat_int(power_power_int(one_one_int),Nb) = one_one_int ).

% power_one
tff(fact_103_power__one,axiom,
    ! [Nb: nat] : aa_nat_complex(power_power_complex(one_one_complex),Nb) = one_one_complex ).

% power_one
tff(fact_104_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_105_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_106_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_107_power__one__right,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),one_one_nat) = A ).

% power_one_right
tff(fact_108_semiring__norm_I78_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(78)
tff(fact_109_semiring__norm_I71_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(71)
tff(fact_110_semiring__norm_I75_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),one) ).

% semiring_norm(75)
tff(fact_111_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_112_numeral__eq__one__iff,axiom,
    ! [Nb: num] :
      ( ( numera6690914467698888265omplex(Nb) = one_one_complex )
    <=> ( Nb = one ) ) ).

% numeral_eq_one_iff
tff(fact_113_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_114_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_115_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_116_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_117_one__eq__numeral__iff,axiom,
    ! [Nb: num] :
      ( ( one_one_complex = numera6690914467698888265omplex(Nb) )
    <=> ( one = Nb ) ) ).

% one_eq_numeral_iff
tff(fact_118_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_119_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_120_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_121_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_122_power__inject__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_123_power__inject__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( ( aa_nat_rat(power_power_rat(A),Mb) = aa_nat_rat(power_power_rat(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_124_power__inject__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_125_power__inject__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),Mb) = aa_nat_int(power_power_int(A),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% power_inject_exp
tff(fact_126_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_127_semiring__norm_I69_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),one) ).

% semiring_norm(69)
tff(fact_128__C5_Ohyps_C_I5_J,axiom,
    m = aa_nat_nat(suc,na) ).

% "5.hyps"(5)
tff(fact_129_numeral__plus__one,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Nb)),one_one_complex) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_130_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(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_131_numeral__plus__one,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Nb)),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_132_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(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_133_numeral__plus__one,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Nb),one)) ).

% numeral_plus_one
tff(fact_134_one__plus__numeral,axiom,
    ! [Nb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_135_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(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_136_one__plus__numeral,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_137_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(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_138_one__plus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% one_plus_numeral
tff(fact_139_power__strict__increasing__iff,axiom,
    ! [B2: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_140_power__strict__increasing__iff,axiom,
    ! [B2: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),Xb)),aa_nat_rat(power_power_rat(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_141_power__strict__increasing__iff,axiom,
    ! [B2: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_142_power__strict__increasing__iff,axiom,
    ! [B2: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),Xb)),aa_nat_int(power_power_int(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_143_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_144_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_145_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_146_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_147_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% one_add_one
tff(fact_148_power__increasing__iff,axiom,
    ! [B2: real,Xb: nat,Y: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_149_power__increasing__iff,axiom,
    ! [B2: rat,Xb: nat,Y: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),Xb)),aa_nat_rat(power_power_rat(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_150_power__increasing__iff,axiom,
    ! [B2: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_151_power__increasing__iff,axiom,
    ! [B2: int,Xb: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),Xb)),aa_nat_int(power_power_int(B2),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power_increasing_iff
tff(fact_152_add__self__div__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Mb ).

% add_self_div_2
tff(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_is__num__normalize_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ).

% is_num_normalize(1)
tff(fact_162_is__num__normalize_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% is_num_normalize(1)
tff(fact_163_is__num__normalize_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% is_num_normalize(1)
tff(fact_164_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_complex_complex(plus_plus_complex(one_one_complex),numera6690914467698888265omplex(Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% one_plus_numeral_commute
tff(fact_165_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_num_real(numeral_numeral_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% one_plus_numeral_commute
tff(fact_166_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_num_rat(numeral_numeral_rat,Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_167_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_168_one__plus__numeral__commute,axiom,
    ! [Xb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_num_int(numeral_numeral_int,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_169_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),one)
    <=> ( Xb = one ) ) ).

% le_num_One_iff
tff(fact_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_Bit0
tff(fact_189_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_190_numeral__Bit0,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_Bit0
tff(fact_191_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_strict_increasing
tff(fact_192_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_strict_increasing
tff(fact_193_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_strict_increasing
tff(fact_194_power__strict__increasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_strict_increasing
tff(fact_195_power__less__imp__less__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_196_power__less__imp__less__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_197_power__less__imp__less__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_198_power__less__imp__less__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% power_less_imp_less_exp
tff(fact_199_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_increasing
tff(fact_200_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_increasing
tff(fact_201_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_increasing
tff(fact_202_power__increasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)) ) ) ).

% power_increasing
tff(fact_203_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_204_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_205_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_206_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_207_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_208_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_209_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_210_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_211_numeral__One,axiom,
    numera6690914467698888265omplex(one) = one_one_complex ).

% numeral_One
tff(fact_212_numeral__One,axiom,
    aa_num_real(numeral_numeral_real,one) = one_one_real ).

% numeral_One
tff(fact_213_numeral__One,axiom,
    aa_num_rat(numeral_numeral_rat,one) = one_one_rat ).

% numeral_One
tff(fact_214_numeral__One,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numeral_One
tff(fact_215_numeral__One,axiom,
    aa_num_int(numeral_numeral_int,one) = one_one_int ).

% numeral_One
tff(fact_216_numerals_I1_J,axiom,
    aa_num_nat(numeral_numeral_nat,one) = one_one_nat ).

% numerals(1)
tff(fact_217_ex__power__ivl1,axiom,
    ! [B2: 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))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),K)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N2)),K)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_218_ex__power__ivl2,axiom,
    ! [B2: 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))),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),N2)),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,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_219_power2__commute,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_220_power2__commute,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_221_power2__commute,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),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),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_222_power2__commute,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_commute
tff(fact_223_power__le__imp__le__exp,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_224_power__le__imp__le__exp,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_225_power__le__imp__le__exp,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_226_power__le__imp__le__exp,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_le_imp_le_exp
tff(fact_227_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_228_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_229_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_230_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_231_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_232_diff__le__diff__pow,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Nb))) ) ).

% diff_le_diff_pow
tff(fact_233_power__divide,axiom,
    ! [A: complex,B2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(divide1717551699836669952omplex(A),B2)),Nb) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B2),Nb)) ).

% power_divide
tff(fact_234_power__divide,axiom,
    ! [A: real,B2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),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,B2),Nb)) ).

% power_divide
tff(fact_235_power__divide,axiom,
    ! [A: rat,B2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),B2)),Nb) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ).

% power_divide
tff(fact_236_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_237_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_238_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_239__092_060open_062high_Ax_An_A_060_A2_A_094_Am_A_092_060and_062_Alow_Ax_An_A_060_A2_A_094_An_092_060close_062,axiom,
    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),na)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),na)) ) ).

% \<open>high x n < 2 ^ m \<and> low x n < 2 ^ n\<close>
tff(fact_240_le__add__diff__inverse,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_241_le__add__diff__inverse,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(minus_minus_rat(A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_242_le__add__diff__inverse,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_243_le__add__diff__inverse,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% le_add_diff_inverse
tff(fact_244_le__add__diff__inverse2,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_245_le__add__diff__inverse2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_246_le__add__diff__inverse2,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_247_le__add__diff__inverse2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ) ) ).

% le_add_diff_inverse2
tff(fact_248_div__exp__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_249_div__exp__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ).

% div_exp_eq
tff(fact_250_field__less__half__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_251_field__less__half__sum,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ).

% field_less_half_sum
tff(fact_252_high__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Y ) ) ).

% high_inv
tff(fact_253_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ).

% diff_diff_left
tff(fact_254_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_255_nat__add__left__cancel__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),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,Mb),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_256_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),N2)
     => ~ ! [N2: nat] : Xb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_nat_nat(suc,N2)) ) ).

% even_odd_cases
tff(fact_257_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_bit_concat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),D)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),D)),D) = Xb ).

% bit_split_inv
tff(fact_258_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( aa_nat_nat(suc,X2) = aa_nat_nat(suc,Y2) )
    <=> ( X2 = Y2 ) ) ).

% nat.inject
tff(fact_259_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_260_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb)) = numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_261_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_262_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb)) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_263_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) = aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_264_numeral__times__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb)) ).

% numeral_times_numeral
tff(fact_265_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Zb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),Zb)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(times_times_num(V),W))),Zb) ).

% mult_numeral_left_semiring_numeral
tff(fact_266_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Zb: 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)),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(V),W))),Zb) ).

% mult_numeral_left_semiring_numeral
tff(fact_267_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Zb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Zb) ).

% mult_numeral_left_semiring_numeral
tff(fact_268_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Zb: 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)),Zb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(V),W))),Zb) ).

% mult_numeral_left_semiring_numeral
tff(fact_269_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Zb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Zb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Zb) ).

% mult_numeral_left_semiring_numeral
tff(fact_270_low__inv,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),Xb)),Nb) = Xb ) ) ).

% low_inv
tff(fact_271_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_272_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_273_div__by__1,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),one_one_complex) = A ).

% div_by_1
tff(fact_274_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_275_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),one_one_rat) = A ).

% div_by_1
tff(fact_276_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_277_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_278_lessI,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Nb)) ).

% lessI
tff(fact_279_Suc__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) ) ).

% Suc_mono
tff(fact_280_Suc__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_eq
tff(fact_281_Suc__le__mono,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% Suc_le_mono
tff(fact_282_add__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc_right
tff(fact_283_nat__add__left__cancel__less,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_284_diff__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% diff_Suc_Suc
tff(fact_285_Suc__diff__diff,axiom,
    ! [Mb: 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,Mb)),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,Mb),Nb)),K) ).

% Suc_diff_diff
tff(fact_286_nat__1__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_287_nat__mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = one_one_nat )
    <=> ( ( Mb = one_one_nat )
        & ( Nb = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_288_semiring__norm_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(6)
tff(fact_289_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_290_distrib__left__numeral,axiom,
    ! [V: num,B2: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(plus_plus_complex(B2),C)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% distrib_left_numeral
tff(fact_291_distrib__left__numeral,axiom,
    ! [V: num,B2: 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,B2),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)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% distrib_left_numeral
tff(fact_292_distrib__left__numeral,axiom,
    ! [V: num,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).

% distrib_left_numeral
tff(fact_293_distrib__left__numeral,axiom,
    ! [V: num,B2: 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,B2),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)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,V)),C)) ).

% distrib_left_numeral
tff(fact_294_distrib__left__numeral,axiom,
    ! [V: num,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% distrib_left_numeral
tff(fact_295_distrib__right__numeral,axiom,
    ! [A: complex,B2: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B2),numera6690914467698888265omplex(V))) ).

% distrib_right_numeral
tff(fact_296_distrib__right__numeral,axiom,
    ! [A: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),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,B2),aa_num_real(numeral_numeral_real,V))) ).

% distrib_right_numeral
tff(fact_297_distrib__right__numeral,axiom,
    ! [A: rat,B2: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_num_rat(numeral_numeral_rat,V))) ).

% distrib_right_numeral
tff(fact_298_distrib__right__numeral,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),aa_num_nat(numeral_numeral_nat,V))) ).

% distrib_right_numeral
tff(fact_299_distrib__right__numeral,axiom,
    ! [A: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% distrib_right_numeral
tff(fact_300_left__diff__distrib__numeral,axiom,
    ! [A: complex,B2: complex,V: num] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),B2)),numera6690914467698888265omplex(V)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(V))),aa_complex_complex(times_times_complex(B2),numera6690914467698888265omplex(V))) ).

% left_diff_distrib_numeral
tff(fact_301_left__diff__distrib__numeral,axiom,
    ! [A: real,B2: real,V: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),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,B2),aa_num_real(numeral_numeral_real,V))) ).

% left_diff_distrib_numeral
tff(fact_302_left__diff__distrib__numeral,axiom,
    ! [A: rat,B2: rat,V: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B2)),aa_num_rat(numeral_numeral_rat,V)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_num_rat(numeral_numeral_rat,V))) ).

% left_diff_distrib_numeral
tff(fact_303_left__diff__distrib__numeral,axiom,
    ! [A: int,B2: int,V: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),aa_num_int(numeral_numeral_int,V)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_num_int(numeral_numeral_int,V))) ).

% left_diff_distrib_numeral
tff(fact_304_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: complex,C: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),aa_complex_complex(minus_minus_complex(B2),C)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),B2)),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(V)),C)) ).

% right_diff_distrib_numeral
tff(fact_305_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: 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,B2),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)),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_306_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(minus_minus_rat(B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_307_right__diff__distrib__numeral,axiom,
    ! [V: num,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),C)) ).

% right_diff_distrib_numeral
tff(fact_308_mult__Suc__right,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc_right
tff(fact_309_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_310_semiring__norm_I2_J,axiom,
    aa_num_num(plus_plus_num(one),one) = aa_num_num(bit0,one) ).

% semiring_norm(2)
tff(fact_311_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: 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,B2),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))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_312_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B2) ) ).

% le_divide_eq_numeral1(1)
tff(fact_313_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: 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,B2),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_314_divide__le__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_315_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_316_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))),B2) ) ).

% less_divide_eq_numeral1(1)
tff(fact_317_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: 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,B2),aa_num_real(numeral_numeral_real,W))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_318_divide__less__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W))),A)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_319_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))),I) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,J)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_320_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_321_power__add__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_322_power__add__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_323_power__add__numeral,axiom,
    ! [A: rat,Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_324_power__add__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_325_power__add__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% power_add_numeral
tff(fact_326_power__add__numeral2,axiom,
    ! [A: complex,Mb: num,Nb: num,B2: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B2) ).

% power_add_numeral2
tff(fact_327_power__add__numeral2,axiom,
    ! [A: real,Mb: num,Nb: num,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = 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(plus_plus_num(Mb),Nb)))),B2) ).

% power_add_numeral2
tff(fact_328_power__add__numeral2,axiom,
    ! [A: rat,Mb: num,Nb: num,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B2) ).

% power_add_numeral2
tff(fact_329_power__add__numeral2,axiom,
    ! [A: nat,Mb: num,Nb: num,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = 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(plus_plus_num(Mb),Nb)))),B2) ).

% power_add_numeral2
tff(fact_330_power__add__numeral2,axiom,
    ! [A: int,Mb: num,Nb: num,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Nb))),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(plus_plus_num(Mb),Nb)))),B2) ).

% power_add_numeral2
tff(fact_331_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(plus_plus_num(Nb),one)) ).

% Suc_numeral
tff(fact_332_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_333_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_334_div2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% div2_Suc_Suc
tff(fact_335_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_336_Suc__mult__le__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),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,Mb),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_337_Suc__mult__less__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb)),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,Mb),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_338_mult__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% mult_Suc
tff(fact_339_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(suc,Xb) = aa_nat_nat(suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_340_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa_nat_nat(suc,Nb) ).

% n_not_Suc_n
tff(fact_341_Suc__mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,K)),Nb) )
    <=> ( Mb = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_342_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_343_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_344_power__Suc2,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) ).

% power_Suc2
tff(fact_345_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_346_power__Suc2,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) ).

% power_Suc2
tff(fact_347_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_348_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_349_power__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_Suc
tff(fact_350_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_351_power__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_Suc
tff(fact_352_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),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,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_353_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_354_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% ring_class.ring_distribs(2)
tff(fact_355_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_356_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_357_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_358_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_359_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_360_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_361_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% comm_semiring_class.distrib
tff(fact_362_distrib__left,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% distrib_left
tff(fact_363_distrib__left,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_364_distrib__left,axiom,
    ! [A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_365_distrib__left,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_366_distrib__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),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,B2),C)) ).

% distrib_right
tff(fact_367_distrib__right,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% distrib_right
tff(fact_368_distrib__right,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),C)) ).

% distrib_right
tff(fact_369_distrib__right,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% distrib_right
tff(fact_370_combine__common__factor,axiom,
    ! [A: real,E: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),C)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),E)),C) ).

% combine_common_factor
tff(fact_371_combine__common__factor,axiom,
    ! [A: rat,E: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),E)),C) ).

% combine_common_factor
tff(fact_372_combine__common__factor,axiom,
    ! [A: nat,E: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),E)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),E)),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),E)),C) ).

% combine_common_factor
tff(fact_373_combine__common__factor,axiom,
    ! [A: int,E: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),E)),C) ).

% combine_common_factor
tff(fact_374_left__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),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,B2),C)) ).

% left_diff_distrib
tff(fact_375_left__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B2)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% left_diff_distrib
tff(fact_376_left__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% left_diff_distrib
tff(fact_377_right__diff__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib
tff(fact_378_right__diff__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(minus_minus_rat(B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib
tff(fact_379_right__diff__distrib,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_380_left__diff__distrib_H,axiom,
    ! [B2: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)) ).

% left_diff_distrib'
tff(fact_381_left__diff__distrib_H,axiom,
    ! [B2: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B2),C)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)) ).

% left_diff_distrib'
tff(fact_382_left__diff__distrib_H,axiom,
    ! [B2: 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,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_383_left__diff__distrib_H,axiom,
    ! [B2: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_384_right__diff__distrib_H,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ).

% right_diff_distrib'
tff(fact_385_right__diff__distrib_H,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(minus_minus_rat(B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% right_diff_distrib'
tff(fact_386_right__diff__distrib_H,axiom,
    ! [A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_387_right__diff__distrib_H,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_388_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K)
     => ( ( K != aa_nat_nat(suc,I) )
       => ~ ! [J2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
             => ( K != aa_nat_nat(suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_389_Suc__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_lessD
tff(fact_390_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J2)
           => ( K != aa_nat_nat(suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_391_Suc__lessI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( ( aa_nat_nat(suc,Mb) != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),Nb) ) ) ).

% Suc_lessI
tff(fact_392_less__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( Mb = Nb ) ) ) ).

% less_SucE
tff(fact_393_less__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% less_SucI
tff(fact_394_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_395_less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% less_Suc_eq
tff(fact_396_not__less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb)) ) ).

% not_less_eq
tff(fact_397_All__less__Suc,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,Nb)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,I2) ) ) ) ).

% All_less_Suc
tff(fact_398_Suc__less__eq2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb)
    <=> ? [M: nat] :
          ( ( Mb = aa_nat_nat(suc,M) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),M) ) ) ).

% Suc_less_eq2
tff(fact_399_less__antisym,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
       => ( Mb = Nb ) ) ) ).

% less_antisym
tff(fact_400_Suc__less__SucD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_less_SucD
tff(fact_401_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_402_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I3),aa_nat_nat(suc,I3))
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),K2)
               => ( aa_nat_o(aa_nat_fun_nat_o(P,I3),J2)
                 => ( aa_nat_o(aa_nat_fun_nat_o(P,J2),K2)
                   => aa_nat_o(aa_nat_fun_nat_o(P,I3),K2) ) ) ) )
         => aa_nat_o(aa_nat_fun_nat_o(P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_403_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa_nat_nat(suc,I3) )
           => aa_nat_o(P,I3) )
       => ( ! [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J)
             => ( aa_nat_o(P,aa_nat_nat(suc,I3))
               => aa_nat_o(P,I3) ) )
         => aa_nat_o(P,I) ) ) ) ).

% strict_inc_induct
tff(fact_404_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% not_less_less_Suc_eq
tff(fact_405_Suc__leD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% Suc_leD
tff(fact_406_le__SucE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_SucE
tff(fact_407_le__SucI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_SucI
tff(fact_408_Suc__le__D,axiom,
    ! [Nb: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),M2)
     => ? [M3: nat] : M2 = aa_nat_nat(suc,M3) ) ).

% Suc_le_D
tff(fact_409_le__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        | ( Mb = aa_nat_nat(suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_410_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_411_not__less__eq__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% not_less_eq_eq
tff(fact_412_full__nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,M4)),N2)
             => aa_nat_o(P,M4) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Nb) ) ).

% full_nat_induct
tff(fact_413_nat__induct__at__least,axiom,
    ! [Mb: nat,Nb: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(P,Mb)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_414_transitive__stepwise__le,axiom,
    ! [Mb: nat,Nb: nat,R: fun_nat_fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ! [X3: nat] : aa_nat_o(aa_nat_fun_nat_o(R,X3),X3)
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X3),Y3)
             => ( aa_nat_o(aa_nat_fun_nat_o(R,Y3),Z)
               => aa_nat_o(aa_nat_fun_nat_o(R,X3),Z) ) )
         => ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(R,N2),aa_nat_nat(suc,N2))
           => aa_nat_o(aa_nat_fun_nat_o(R,Mb),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_415_le__cube,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb))) ).

% le_cube
tff(fact_416_le__square,axiom,
    ! [Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Mb)) ).

% le_square
tff(fact_417_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),L)) ) ) ).

% mult_le_mono
tff(fact_418_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ).

% mult_le_mono1
tff(fact_419_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ).

% mult_le_mono2
tff(fact_420_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(suc,A3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(suc,A)) ) ) ).

% nat_arith.suc1
tff(fact_421_add__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% add_Suc
tff(fact_422_add__Suc__shift,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(suc,Nb)) ).

% add_Suc_shift
tff(fact_423_add__mult__distrib,axiom,
    ! [Mb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% add_mult_distrib
tff(fact_424_add__mult__distrib2,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% add_mult_distrib2
tff(fact_425_zero__induct__lemma,axiom,
    ! [P: fun_nat_o,K: nat,I: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_426_diff__mult__distrib,axiom,
    ! [Mb: 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,Mb),Nb)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)) ).

% diff_mult_distrib
tff(fact_427_diff__mult__distrib2,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% diff_mult_distrib2
tff(fact_428_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_429_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_430_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_431_add__One__commute,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),Nb) = aa_num_num(plus_plus_num(Nb),one) ).

% add_One_commute
tff(fact_432_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_433_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_434_power__commutes,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_commutes
tff(fact_435_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_436_power__commutes,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb)) ).

% power_commutes
tff(fact_437_power__mult__distrib,axiom,
    ! [A: complex,B2: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(times_times_complex(A),B2)),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Nb)),aa_nat_complex(power_power_complex(B2),Nb)) ).

% power_mult_distrib
tff(fact_438_power__mult__distrib,axiom,
    ! [A: real,B2: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),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,B2),Nb)) ).

% power_mult_distrib
tff(fact_439_power__mult__distrib,axiom,
    ! [A: rat,B2: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_nat_rat(power_power_rat(B2),Nb)) ).

% power_mult_distrib
tff(fact_440_power__mult__distrib,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),Nb)) ).

% power_mult_distrib
tff(fact_441_power__mult__distrib,axiom,
    ! [A: int,B2: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B2),Nb)) ).

% power_mult_distrib
tff(fact_442_power__commuting__commutes,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(power_power_complex(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_443_power__commuting__commutes,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_444_power__commuting__commutes,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(power_power_rat(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_445_power__commuting__commutes,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),Xb) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_446_power__commuting__commutes,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(Xb),Nb)) ) ) ).

% power_commuting_commutes
tff(fact_447_power__mult,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),Nb) ).

% power_mult
tff(fact_448_power__mult,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),Nb) ).

% power_mult
tff(fact_449_power__mult,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Mb)),Nb) ).

% power_mult
tff(fact_450_power__mult,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),Mb)),Nb) ).

% power_mult
tff(fact_451_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),K)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_452_div__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Q2) ).

% div_mult2_eq
tff(fact_453_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_454_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_455_power__odd__eq,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(aa_nat_rat(power_power_rat(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_456_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_457_power__odd__eq,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),Nb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% power_odd_eq
tff(fact_458_less__1__mult,axiom,
    ! [Mb: real,Nb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_459_less__1__mult,axiom,
    ! [Mb: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Mb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_460_less__1__mult,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_461_less__1__mult,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) ) ) ).

% less_1_mult
tff(fact_462_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_463_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B2)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_464_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E)),C) = D ) ) ).

% eq_add_iff1
tff(fact_465_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D) )
    <=> ( C = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_466_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B2),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_467_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D) )
    <=> ( C = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E)),D) ) ) ).

% eq_add_iff2
tff(fact_468_square__diff__square__factored,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% square_diff_square_factored
tff(fact_469_square__diff__square__factored,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),aa_rat_rat(minus_minus_rat(Xb),Y)) ).

% square_diff_square_factored
tff(fact_470_square__diff__square__factored,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) ).

% square_diff_square_factored
tff(fact_471_mult__diff__mult,axiom,
    ! [Xb: real,Y: real,A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A)),B2)) ).

% mult_diff_mult
tff(fact_472_mult__diff__mult,axiom,
    ! [Xb: rat,Y: rat,A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(minus_minus_rat(Y),B2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(Xb),A)),B2)) ).

% mult_diff_mult
tff(fact_473_mult__diff__mult,axiom,
    ! [Xb: int,Y: int,A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A)),B2)) ).

% mult_diff_mult
tff(fact_474_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_real,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_475_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_476_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_477_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_478_lift__Suc__mono__less,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N3)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_less
tff(fact_479_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_real,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(F,Nb)),aa_nat_real(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_480_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_rat,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(F,Nb)),aa_nat_rat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_481_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_num,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_nat_num(F,Nb)),aa_nat_num(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_482_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_nat,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,Nb)),aa_nat_nat(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_483_lift__Suc__mono__less__iff,axiom,
    ! [F: fun_nat_int,Nb: nat,Mb: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(F,Nb)),aa_nat_int(F,Mb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_484_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : ord_less_eq_set_nat(aa_nat_set_nat(F,N2),aa_nat_set_nat(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_nat(aa_nat_set_nat(F,Nb),aa_nat_set_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_485_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,N2)),aa_nat_rat(F,aa_nat_nat(suc,N2)))
     => ( 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,aa_nat_rat(F,Nb)),aa_nat_rat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_486_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N2)),aa_nat_num(F,aa_nat_nat(suc,N2)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,Nb)),aa_nat_num(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_487_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,N2)),aa_nat_nat(F,aa_nat_nat(suc,N2)))
     => ( 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,aa_nat_nat(F,Nb)),aa_nat_nat(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_488_lift__Suc__mono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,N2)),aa_nat_int(F,aa_nat_nat(suc,N2)))
     => ( 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,aa_nat_int(F,Nb)),aa_nat_int(F,N3)) ) ) ).

% lift_Suc_mono_le
tff(fact_489_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_set_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : ord_less_eq_set_nat(aa_nat_set_nat(F,aa_nat_nat(suc,N2)),aa_nat_set_nat(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => ord_less_eq_set_nat(aa_nat_set_nat(F,N3),aa_nat_set_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_490_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_rat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,aa_nat_nat(suc,N2))),aa_nat_rat(F,N2))
     => ( 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,aa_nat_rat(F,N3)),aa_nat_rat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_491_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_num,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,aa_nat_nat(suc,N2))),aa_nat_num(F,N2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N3)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,N3)),aa_nat_num(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_492_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_nat,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,aa_nat_nat(suc,N2))),aa_nat_nat(F,N2))
     => ( 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,aa_nat_nat(F,N3)),aa_nat_nat(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_493_lift__Suc__antimono__le,axiom,
    ! [F: fun_nat_int,Nb: nat,N3: nat] :
      ( ! [N2: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,aa_nat_nat(suc,N2))),aa_nat_int(F,N2))
     => ( 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,aa_nat_int(F,N3)),aa_nat_int(F,Nb)) ) ) ).

% lift_Suc_antimono_le
tff(fact_494_Suc__leI,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb) ) ).

% Suc_leI
tff(fact_495_Suc__le__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_eq
tff(fact_496_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,I)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) )
         => aa_nat_o(P,J) ) ) ) ).

% dec_induct
tff(fact_497_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(P,J)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),N2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),J)
               => ( aa_nat_o(P,aa_nat_nat(suc,N2))
                 => aa_nat_o(P,N2) ) ) )
         => aa_nat_o(P,I) ) ) ) ).

% inc_induct
tff(fact_498_Suc__le__lessD,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% Suc_le_lessD
tff(fact_499_le__less__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
      <=> ( Nb = Mb ) ) ) ).

% le_less_Suc_eq
tff(fact_500_less__Suc__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_Suc_eq_le
tff(fact_501_less__eq__Suc__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Nb)),Mb) ) ).

% less_eq_Suc_le
tff(fact_502_le__imp__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_503_less__imp__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ? [K2: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K2)) ) ).

% less_imp_Suc_add
tff(fact_504_less__iff__Suc__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3)) ) ).

% less_iff_Suc_add
tff(fact_505_less__add__Suc2,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),I))) ).

% less_add_Suc2
tff(fact_506_less__add__Suc1,axiom,
    ! [I: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb))) ).

% less_add_Suc1
tff(fact_507_less__natE,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ~ ! [Q3: nat] : Nb != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q3)) ) ).

% less_natE
tff(fact_508_diff__less__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(suc,Mb)) ).

% diff_less_Suc
tff(fact_509_Suc__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_510_Suc__diff__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ) ) ).

% Suc_diff_le
tff(fact_511_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_512_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_513_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_514_diff__Suc__eq__diff__pred,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_515_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(plus_plus_num(V),one))),Nb) ).

% Suc_nat_number_of_add
tff(fact_516_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q2)))
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_517_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_518_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B2),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_519_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E)),D)) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_520_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_521_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B2)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_522_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E)),C)),D) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_523_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_524_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(B2),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_525_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),E)),D)) ) ).

% less_add_iff2
tff(fact_526_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B2: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),E)),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),E)),D))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_527_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B2: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),E)),D))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),B2)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_528_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B2: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),E)),D))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),E)),C)),D) ) ).

% less_add_iff1
tff(fact_529_square__diff__one__factored,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Xb)),one_one_complex) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Xb),one_one_complex)),aa_complex_complex(minus_minus_complex(Xb),one_one_complex)) ).

% square_diff_one_factored
tff(fact_530_square__diff__one__factored,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),one_one_real) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ).

% square_diff_one_factored
tff(fact_531_square__diff__one__factored,axiom,
    ! [Xb: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)),aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) ).

% square_diff_one_factored
tff(fact_532_square__diff__one__factored,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),one_one_int) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),one_one_int)) ).

% square_diff_one_factored
tff(fact_533_mult__numeral__1__right,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(one)) = A ).

% mult_numeral_1_right
tff(fact_534_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_535_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,one)) = A ).

% mult_numeral_1_right
tff(fact_536_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_537_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,one)) = A ).

% mult_numeral_1_right
tff(fact_538_mult__numeral__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(one)),A) = A ).

% mult_numeral_1
tff(fact_539_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_540_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,one)),A) = A ).

% mult_numeral_1
tff(fact_541_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_542_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,one)),A) = A ).

% mult_numeral_1
tff(fact_543_left__right__inverse__power,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = one_one_complex )
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) = one_one_complex ) ) ).

% left_right_inverse_power
tff(fact_544_left__right__inverse__power,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = one_one_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) = one_one_real ) ) ).

% left_right_inverse_power
tff(fact_545_left__right__inverse__power,axiom,
    ! [Xb: rat,Y: rat,Nb: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(Y),Nb)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_546_left__right__inverse__power,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = one_one_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_547_left__right__inverse__power,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_548_power__add,axiom,
    ! [A: complex,Mb: nat,Nb: nat] : aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ).

% power_add
tff(fact_549_power__add,axiom,
    ! [A: real,Mb: nat,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_add
tff(fact_550_power__add,axiom,
    ! [A: rat,Mb: nat,Nb: nat] : aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_add
tff(fact_551_power__add,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ).

% power_add
tff(fact_552_power__add,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_add
tff(fact_553_Suc__div__le__mono,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,Mb)),Nb)) ).

% Suc_div_le_mono
tff(fact_554_less__mult__imp__div__less,axiom,
    ! [Mb: nat,I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_555_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))),Mb) ).

% times_div_less_eq_dividend
tff(fact_556_div__times__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)),Mb) ).

% div_times_less_eq_dividend
tff(fact_557_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_558_power__gt1__lemma,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_559_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_560_power__gt1__lemma,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_gt1_lemma
tff(fact_561_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_562_power__less__power__Suc,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(A),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_563_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_564_power__less__power__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))) ) ).

% power_less_power_Suc
tff(fact_565_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_566_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_567_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_568_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_569_linorder__neqE__linordered__idom,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_570_linorder__neqE__linordered__idom,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_571_linorder__neqE__linordered__idom,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_572_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_573_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_574_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_575_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_576_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_577_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_578_nat__neq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb != Nb )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ).

% nat_neq_iff
tff(fact_579_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_not_refl
tff(fact_580_less__not__refl2,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( Mb != Nb ) ) ).

% less_not_refl2
tff(fact_581_less__not__refl3,axiom,
    ! [S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Ta)
     => ( S != Ta ) ) ).

% less_not_refl3
tff(fact_582_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_583_nat__less__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
             => aa_nat_o(P,M4) )
         => aa_nat_o(P,N2) )
     => aa_nat_o(P,Nb) ) ).

% nat_less_induct
tff(fact_584_infinite__descent,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(P,N2)
         => ? [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
              & ~ aa_nat_o(P,M4) ) )
     => aa_nat_o(P,Nb) ) ).

% infinite_descent
tff(fact_585_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_586_le__refl,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Nb) ).

% le_refl
tff(fact_587_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),K) ) ) ).

% le_trans
tff(fact_588_eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = Nb )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% eq_imp_le
tff(fact_589_le__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% le_antisym
tff(fact_590_nat__le__linear,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ).

% nat_le_linear
tff(fact_591_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: 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),B2) )
       => ? [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_592_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),K)),J) ).

% diff_commute
tff(fact_593_left__add__twice,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),A)),B2) ).

% left_add_twice
tff(fact_594_left__add__twice,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = 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)),B2) ).

% left_add_twice
tff(fact_595_left__add__twice,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),A)),B2) ).

% left_add_twice
tff(fact_596_left__add__twice,axiom,
    ! [A: nat,B2: 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),B2)) = 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)),B2) ).

% left_add_twice
tff(fact_597_left__add__twice,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),B2) ).

% left_add_twice
tff(fact_598_mult__2__right,axiom,
    ! [Zb: complex] : aa_complex_complex(times_times_complex(Zb),numera6690914467698888265omplex(aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(Zb),Zb) ).

% mult_2_right
tff(fact_599_mult__2__right,axiom,
    ! [Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),Zb) ).

% mult_2_right
tff(fact_600_mult__2__right,axiom,
    ! [Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),Zb) ).

% mult_2_right
tff(fact_601_mult__2__right,axiom,
    ! [Zb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Zb),Zb) ).

% mult_2_right
tff(fact_602_mult__2__right,axiom,
    ! [Zb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Zb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),Zb) ).

% mult_2_right
tff(fact_603_mult__2,axiom,
    ! [Zb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Zb) = aa_complex_complex(plus_plus_complex(Zb),Zb) ).

% mult_2
tff(fact_604_mult__2,axiom,
    ! [Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Zb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),Zb) ).

% mult_2
tff(fact_605_mult__2,axiom,
    ! [Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Zb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),Zb) ).

% mult_2
tff(fact_606_mult__2,axiom,
    ! [Zb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Zb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Zb),Zb) ).

% mult_2
tff(fact_607_mult__2,axiom,
    ! [Zb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Zb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),Zb) ).

% mult_2
tff(fact_608_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_609_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_610_power2__eq__square,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% power2_eq_square
tff(fact_611_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_612_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_613_power4__eq__xxxx,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_614_power4__eq__xxxx,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_615_power4__eq__xxxx,axiom,
    ! [Xb: rat] : aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_616_power4__eq__xxxx,axiom,
    ! [Xb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_617_power4__eq__xxxx,axiom,
    ! [Xb: int] : aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),Xb)),Xb) ).

% power4_eq_xxxx
tff(fact_618_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_619_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_620_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_621_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_622_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,I),J)),U)),Mb)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_623_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),U)),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),U)),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I)),U)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_624_power2__sum,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_625_power2__sum,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_626_power2__sum,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_627_power2__sum,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_628_power2__sum,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_sum
tff(fact_629_add__diff__add,axiom,
    ! [A: real,C: real,B2: 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,B2),D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D)) ).

% add_diff_add
tff(fact_630_add__diff__add,axiom,
    ! [A: rat,C: rat,B2: rat,D: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B2)),aa_rat_rat(minus_minus_rat(C),D)) ).

% add_diff_add
tff(fact_631_add__diff__add,axiom,
    ! [A: int,C: int,B2: int,D: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D)) ).

% add_diff_add
tff(fact_632_nat__less__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & ( Mb != Nb ) ) ) ).

% nat_less_le
tff(fact_633_less__imp__le__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_imp_le_nat
tff(fact_634_le__eq__less__or__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_635_less__or__eq__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Mb = Nb ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% less_or_eq_imp_le
tff(fact_636_le__neq__implies__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( ( Mb != Nb )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% le_neq_implies_less
tff(fact_637_less__mono__imp__le__mono,axiom,
    ! [F: fun_nat_nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),J2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,I3)),aa_nat_nat(F,J2)) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,I)),aa_nat_nat(F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_638_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),K) ) ).

% add_lessD1
tff(fact_639_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_less_mono
tff(fact_640_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),I) ).

% not_add_less1
tff(fact_641_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),I) ).

% not_add_less2
tff(fact_642_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_less_mono1
tff(fact_643_trans__less__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_less_add1
tff(fact_644_trans__less__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_less_add2
tff(fact_645_less__add__eq__less,axiom,
    ! [K: nat,L: nat,Mb: 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,Mb),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,Mb),Nb) ) ) ).

% less_add_eq_less
tff(fact_646_add__leE,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ) ).

% add_leE
tff(fact_647_le__add1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) ).

% le_add1
tff(fact_648_le__add2,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% le_add2
tff(fact_649_add__leD1,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% add_leD1
tff(fact_650_add__leD2,axiom,
    ! [Mb: 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,Mb),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb) ) ).

% add_leD2
tff(fact_651_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
     => ? [N2: nat] : L = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),N2) ) ).

% le_Suc_ex
tff(fact_652_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ) ).

% add_le_mono
tff(fact_653_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) ) ).

% add_le_mono1
tff(fact_654_trans__le__add1,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),Mb)) ) ).

% trans_le_add1
tff(fact_655_trans__le__add2,axiom,
    ! [I: nat,J: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),J)) ) ).

% trans_le_add2
tff(fact_656_nat__le__iff__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
    <=> ? [K3: nat] : Nb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K3) ) ).

% nat_le_iff_add
tff(fact_657_diff__less__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),L)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ) ).

% diff_less_mono2
tff(fact_658_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_659_eq__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K) )
        <=> ( Mb = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_660_le__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% le_diff_iff
tff(fact_661_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_662_diff__le__mono,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),L)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_663_diff__le__self,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ).

% diff_le_self
tff(fact_664_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ) ) ).

% le_diff_iff'
tff(fact_665_diff__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,L: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,L),Mb)) ) ).

% diff_le_mono2
tff(fact_666_power2__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(minus_minus_complex(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_667_power2__diff,axiom,
    ! [Xb: real,Y: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_668_power2__diff,axiom,
    ! [Xb: rat,Y: rat] : aa_nat_rat(power_power_rat(aa_rat_rat(minus_minus_rat(Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_669_power2__diff,axiom,
    ! [Xb: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)),Y)) ).

% power2_diff
tff(fact_670_Nat_Odiff__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) ).

% Nat.diff_cancel
tff(fact_671_diff__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ).

% diff_cancel2
tff(fact_672_diff__add__inverse,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),Nb) = Mb ).

% diff_add_inverse
tff(fact_673_diff__add__inverse2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),Nb) = Mb ).

% diff_add_inverse2
tff(fact_674_add__le__add__imp__diff__le,axiom,
    ! [I: 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,I),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,I),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_675_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,Nb: rat,J: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Nb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),K))
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Nb)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Nb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),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_676_add__le__add__imp__diff__le,axiom,
    ! [I: 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,I),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,I),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_677_add__le__add__imp__diff__le,axiom,
    ! [I: int,K: int,Nb: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),K))
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)),J) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_678_add__le__imp__le__diff,axiom,
    ! [I: 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,I),K)),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_679_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,Nb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),aa_rat_rat(minus_minus_rat(Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_680_add__le__imp__le__diff,axiom,
    ! [I: 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,I),K)),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_681_add__le__imp__le__diff,axiom,
    ! [I: int,K: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),aa_int_int(aa_int_fun_int_int(minus_minus_int,Nb),K)) ) ).

% add_le_imp_le_diff
tff(fact_682_add__mono1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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,B2),one_one_real)) ) ).

% add_mono1
tff(fact_683_add__mono1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),one_one_rat)) ) ).

% add_mono1
tff(fact_684_add__mono1,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => 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,B2),one_one_nat)) ) ).

% add_mono1
tff(fact_685_add__mono1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)) ) ).

% add_mono1
tff(fact_686_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_687_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_688_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_689_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_690_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_691_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(minus_minus_rat(A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_692_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_693_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = A ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_694_mono__nat__linear__lb,axiom,
    ! [F: fun_nat_nat,Mb: nat,K: nat] :
      ( ! [M3: nat,N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,M3)),aa_nat_nat(F,N2)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(F,Mb)),K)),aa_nat_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))) ) ).

% mono_nat_linear_lb
tff(fact_695_less__diff__iff,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( 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,Mb),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,Mb),Nb) ) ) ) ).

% less_diff_iff
tff(fact_696_diff__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),C)) ) ) ).

% diff_less_mono
tff(fact_697_add__diff__inverse__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = Mb ) ) ).

% add_diff_inverse_nat
tff(fact_698_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ).

% less_diff_conv
tff(fact_699_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) = K )
      <=> ( J = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_700_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),I)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_701_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_702_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_703_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ).

% le_diff_conv
tff(fact_704_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),K)),I)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)) ) ) ).

% less_diff_conv2
tff(fact_705_field__sum__of__halves,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_706_field__sum__of__halves,axiom,
    ! [Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(divide_divide_rat(Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) = Xb ).

% field_sum_of_halves
tff(fact_707__092_060open_062res_A_061_A2_A_094_A_Ideg_Adiv_A2_J_A_K_Asc_A_L_Aminy_092_060close_062,axiom,
    res = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),sc)),miny) ).

% \<open>res = 2 ^ (deg div 2) * sc + miny\<close>
tff(fact_708_sum__squares__bound,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_709_sum__squares__bound,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb)),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_squares_bound
tff(fact_710_enat__ord__number_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(1)
tff(fact_711_enat__ord__number_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),numera1916890842035813515d_enat(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ) ).

% enat_ord_number(2)
tff(fact_712_double__not__eq__Suc__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_713_Suc__double__not__eq__double,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ).

% Suc_double_not_eq_double
tff(fact_714_add__diff__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_715_add__diff__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_716_add__diff__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel
tff(fact_717_diff__add__cancel,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_718_diff__add__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_719_diff__add__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = A ).

% diff_add_cancel
tff(fact_720_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ).

% add_diff_cancel_left
tff(fact_721_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2)) = aa_rat_rat(minus_minus_rat(A),B2) ).

% add_diff_cancel_left
tff(fact_722_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ).

% add_diff_cancel_left
tff(fact_723_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_left
tff(fact_724_add__diff__cancel__left_H,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_725_add__diff__cancel__left_H,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_726_add__diff__cancel__left_H,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_727_add__diff__cancel__left_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),A) = B2 ).

% add_diff_cancel_left'
tff(fact_728_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ).

% add_diff_cancel_right
tff(fact_729_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = aa_rat_rat(minus_minus_rat(A),B2) ).

% add_diff_cancel_right
tff(fact_730_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ).

% add_diff_cancel_right
tff(fact_731_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ).

% add_diff_cancel_right
tff(fact_732_add__diff__cancel__right_H,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_733_add__diff__cancel__right_H,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_734_add__diff__cancel__right_H,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_735_add__diff__cancel__right_H,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = A ).

% add_diff_cancel_right'
tff(fact_736_add__right__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_737_add__right__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_738_add__right__cancel,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_739_add__right__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B2 = C ) ) ).

% add_right_cancel
tff(fact_740_add__left__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_741_add__left__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_742_add__left__cancel,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_743_add__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B2 = C ) ) ).

% add_left_cancel
tff(fact_744_add__le__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_cancel_right
tff(fact_745_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_cancel_right
tff(fact_746_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% add_le_cancel_right
tff(fact_747_add__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_cancel_right
tff(fact_748_add__le__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_cancel_left
tff(fact_749_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_cancel_left
tff(fact_750_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% add_le_cancel_left
tff(fact_751_add__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_cancel_left
tff(fact_752_add__less__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_cancel_right
tff(fact_753_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_cancel_right
tff(fact_754_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_cancel_right
tff(fact_755_add__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_cancel_right
tff(fact_756_add__less__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_cancel_left
tff(fact_757_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_cancel_left
tff(fact_758_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_cancel_left
tff(fact_759_add__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_cancel_left
tff(fact_760_mult__1,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(one_one_complex),A) = A ).

% mult_1
tff(fact_761_mult__1,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,one_one_real),A) = A ).

% mult_1
tff(fact_762_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_763_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_764_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_765_mult_Oright__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.right_neutral
tff(fact_766_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_767_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_768_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_769_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_770_semiring__norm_I13_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(bit0,aa_num_num(times_times_num(Mb),Nb))) ).

% semiring_norm(13)
tff(fact_771_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(one),Nb) = Nb ).

% semiring_norm(12)
tff(fact_772_semiring__norm_I11_J,axiom,
    ! [Mb: num] : aa_num_num(times_times_num(Mb),one) = Mb ).

% semiring_norm(11)
tff(fact_773_num__double,axiom,
    ! [Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,one)),Nb) = aa_num_num(bit0,Nb) ).

% num_double
tff(fact_774_power__mult__numeral,axiom,
    ! [A: nat,Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_775_power__mult__numeral,axiom,
    ! [A: real,Mb: num,Nb: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_776_power__mult__numeral,axiom,
    ! [A: int,Mb: num,Nb: num] : aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_777_power__mult__numeral,axiom,
    ! [A: complex,Mb: num,Nb: num] : aa_nat_complex(power_power_complex(aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,Mb))),aa_num_nat(numeral_numeral_nat,Nb)) = aa_nat_complex(power_power_complex(A),aa_num_nat(numeral_numeral_nat,aa_num_num(times_times_num(Mb),Nb))) ).

% power_mult_numeral
tff(fact_778_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X4: real] : member_real(X4,S2)
     => ( ? [Z2: real] :
          ! [X3: real] :
            ( member_real(X3,S2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z2) )
       => ? [Y3: real] :
            ( ! [X4: real] :
                ( member_real(X4,S2)
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Y3) )
            & ! [Z2: real] :
                ( ! [X3: real] :
                    ( member_real(X3,S2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),Z2) )
               => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),Z2) ) ) ) ) ).

% complete_real
tff(fact_779_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N2)) ) ).

% real_arch_pow
tff(fact_780_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_781_add__diff__assoc__enat,axiom,
    ! [Zb: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Zb),Y)
     => ( plus_p3455044024723400733d_enat(Xb,minus_3235023915231533773d_enat(Y,Zb)) = minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Zb) ) ) ).

% add_diff_assoc_enat
tff(fact_782_enat__less__induct,axiom,
    ! [P: fun_Extended_enat_o,Nb: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M4: extended_enat] :
              ( aa_Extended_enat_o(ord_le72135733267957522d_enat(M4),N2)
             => aa_Extended_enat_o(P,M4) )
         => aa_Extended_enat_o(P,N2) )
     => aa_Extended_enat_o(P,Nb) ) ).

% enat_less_induct
tff(fact_783_four__x__squared,axiom,
    ! [Xb: 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,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% four_x_squared
tff(fact_784_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B2: 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,B2),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,B2),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_785_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(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_786_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,L)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(K),L))) ).

% div_mult2_numeral_eq
tff(fact_787_mult_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),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,B2),C)) ).

% mult.left_commute
tff(fact_788_mult_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% mult.left_commute
tff(fact_789_mult_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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,B2),C)) ).

% mult.left_commute
tff(fact_790_mult_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% mult.left_commute
tff(fact_791_mult_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A) ).

% mult.commute
tff(fact_792_mult_Ocommute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A) ).

% mult.commute
tff(fact_793_mult_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A) ).

% mult.commute
tff(fact_794_mult_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),A) ).

% mult.commute
tff(fact_795_mult_Oassoc,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C)) ).

% mult.assoc
tff(fact_796_mult_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% mult.assoc
tff(fact_797_mult_Oassoc,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% mult.assoc
tff(fact_798_mult_Oassoc,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% mult.assoc
tff(fact_799_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_800_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_801_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_802_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_803_one__reorient,axiom,
    ! [Xb: complex] :
      ( ( one_one_complex = Xb )
    <=> ( Xb = one_one_complex ) ) ).

% one_reorient
tff(fact_804_one__reorient,axiom,
    ! [Xb: real] :
      ( ( one_one_real = Xb )
    <=> ( Xb = one_one_real ) ) ).

% one_reorient
tff(fact_805_one__reorient,axiom,
    ! [Xb: rat] :
      ( ( one_one_rat = Xb )
    <=> ( Xb = one_one_rat ) ) ).

% one_reorient
tff(fact_806_one__reorient,axiom,
    ! [Xb: nat] :
      ( ( one_one_nat = Xb )
    <=> ( Xb = one_one_nat ) ) ).

% one_reorient
tff(fact_807_one__reorient,axiom,
    ! [Xb: int] :
      ( ( one_one_int = Xb )
    <=> ( Xb = one_one_int ) ) ).

% one_reorient
tff(fact_808_add__right__imp__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_809_add__right__imp__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_810_add__right__imp__eq,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_811_add__right__imp__eq,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
tff(fact_812_add__left__imp__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_813_add__left__imp__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_814_add__left__imp__eq,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_815_add__left__imp__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
tff(fact_816_add_Oleft__commute,axiom,
    ! [B2: real,A: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),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,B2),C)) ).

% add.left_commute
tff(fact_817_add_Oleft__commute,axiom,
    ! [B2: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% add.left_commute
tff(fact_818_add_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),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,B2),C)) ).

% add.left_commute
tff(fact_819_add_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% add.left_commute
tff(fact_820_add_Ocommute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) ).

% add.commute
tff(fact_821_add_Ocommute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) ).

% add.commute
tff(fact_822_add_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) ).

% add.commute
tff(fact_823_add_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) ).

% add.commute
tff(fact_824_add_Oright__cancel,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),A) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_825_add_Oright__cancel,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_826_add_Oright__cancel,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B2 = C ) ) ).

% add.right_cancel
tff(fact_827_add_Oleft__cancel,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_828_add_Oleft__cancel,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_829_add_Oleft__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B2 = C ) ) ).

% add.left_cancel
tff(fact_830_add_Oassoc,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ).

% add.assoc
tff(fact_831_add_Oassoc,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% add.assoc
tff(fact_832_add_Oassoc,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% add.assoc
tff(fact_833_add_Oassoc,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% add.assoc
tff(fact_834_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B2: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_835_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B2: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B2) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_836_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B2: nat,A: nat] :
      ( ( B3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B2) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B3) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_837_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B2: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% group_cancel.add2
tff(fact_838_group__cancel_Oadd1,axiom,
    ! [A3: real,K: real,A: real,B2: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A3),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_839_group__cancel_Oadd1,axiom,
    ! [A3: rat,K: rat,A: rat,B2: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A3),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_840_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B2: nat] :
      ( ( A3 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A3),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),B2)) ) ) ).

% group_cancel.add1
tff(fact_841_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B2: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A3),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% group_cancel.add1
tff(fact_842_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_843_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_844_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_845_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_846_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_847_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_848_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_849_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_850_diff__right__commute,axiom,
    ! [A: real,C: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C) ).

% diff_right_commute
tff(fact_851_diff__right__commute,axiom,
    ! [A: rat,C: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B2) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B2)),C) ).

% diff_right_commute
tff(fact_852_diff__right__commute,axiom,
    ! [A: nat,C: nat,B2: 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)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)),C) ).

% diff_right_commute
tff(fact_853_diff__right__commute,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) ).

% diff_right_commute
tff(fact_854_diff__eq__diff__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( ( A = B2 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_855_diff__eq__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( ( A = B2 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_856_diff__eq__diff__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( ( A = B2 )
      <=> ( C = D ) ) ) ).

% diff_eq_diff_eq
tff(fact_857_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_imp_le_right
tff(fact_858_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_imp_le_right
tff(fact_859_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% add_le_imp_le_right
tff(fact_860_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_imp_le_right
tff(fact_861_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% add_le_imp_le_left
tff(fact_862_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% add_le_imp_le_left
tff(fact_863_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% add_le_imp_le_left
tff(fact_864_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% add_le_imp_le_left
tff(fact_865_le__iff__add,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ? [C2: nat] : B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C2) ) ).

% le_iff_add
tff(fact_866_add__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => 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,B2),C)) ) ).

% add_right_mono
tff(fact_867_add__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ) ).

% add_right_mono
tff(fact_868_add__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => 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,B2),C)) ) ).

% add_right_mono
tff(fact_869_add__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ).

% add_right_mono
tff(fact_870_less__eqE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ~ ! [C3: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_871_add__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => 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),B2)) ) ).

% add_left_mono
tff(fact_872_add__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2)) ) ).

% add_left_mono
tff(fact_873_add__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => 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),B2)) ) ).

% add_left_mono
tff(fact_874_add__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% add_left_mono
tff(fact_875_add__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_mono
tff(fact_876_add__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_mono
tff(fact_877_add__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_mono
tff(fact_878_add__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_mono
tff(fact_879_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_880_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_881_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_882_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_883_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_884_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_885_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_886_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_887_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_888_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_889_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_890_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_891_diff__eq__diff__less__eq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_892_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_893_diff__eq__diff__less__eq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% diff_eq_diff_less_eq
tff(fact_894_diff__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => 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,B2),C)) ) ).

% diff_right_mono
tff(fact_895_diff__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => 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(B2),C)) ) ).

% diff_right_mono
tff(fact_896_diff__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% diff_right_mono
tff(fact_897_diff__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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),B2)) ) ).

% diff_left_mono
tff(fact_898_diff__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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),B2)) ) ).

% diff_left_mono
tff(fact_899_diff__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)) ) ).

% diff_left_mono
tff(fact_900_diff__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% diff_mono
tff(fact_901_diff__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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(B2),D)) ) ) ).

% diff_mono
tff(fact_902_diff__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_mono
tff(fact_903_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_imp_less_right
tff(fact_904_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_imp_less_right
tff(fact_905_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_imp_less_right
tff(fact_906_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_imp_less_right
tff(fact_907_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% add_less_imp_less_left
tff(fact_908_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% add_less_imp_less_left
tff(fact_909_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% add_less_imp_less_left
tff(fact_910_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% add_less_imp_less_left
tff(fact_911_add__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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,B2),C)) ) ).

% add_strict_right_mono
tff(fact_912_add__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ) ).

% add_strict_right_mono
tff(fact_913_add__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => 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,B2),C)) ) ).

% add_strict_right_mono
tff(fact_914_add__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ).

% add_strict_right_mono
tff(fact_915_add__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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),B2)) ) ).

% add_strict_left_mono
tff(fact_916_add__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2)) ) ).

% add_strict_left_mono
tff(fact_917_add__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => 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),B2)) ) ).

% add_strict_left_mono
tff(fact_918_add__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% add_strict_left_mono
tff(fact_919_add__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_strict_mono
tff(fact_920_add__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_strict_mono
tff(fact_921_add__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_strict_mono
tff(fact_922_add__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_strict_mono
tff(fact_923_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & ( K = L ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_924_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & ( K = L ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_925_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( K = L ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_926_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & ( K = L ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_927_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( ( I = J )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_928_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_929_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_930_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_931_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_932_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_933_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_934_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_935_diff__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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,B2),C)) ) ).

% diff_strict_right_mono
tff(fact_936_diff__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => 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(B2),C)) ) ).

% diff_strict_right_mono
tff(fact_937_diff__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% diff_strict_right_mono
tff(fact_938_diff__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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),B2)) ) ).

% diff_strict_left_mono
tff(fact_939_diff__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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),B2)) ) ).

% diff_strict_left_mono
tff(fact_940_diff__strict__left__mono,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2)) ) ).

% diff_strict_left_mono
tff(fact_941_diff__eq__diff__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),D) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_942_diff__eq__diff__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(minus_minus_rat(C),D) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_943_diff__eq__diff__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D) ) ) ).

% diff_eq_diff_less
tff(fact_944_diff__strict__mono,axiom,
    ! [A: real,B2: real,D: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_945_diff__strict__mono,axiom,
    ! [A: rat,B2: rat,D: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( 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(B2),D)) ) ) ).

% diff_strict_mono
tff(fact_946_diff__strict__mono,axiom,
    ! [A: int,B2: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)) ) ) ).

% diff_strict_mono
tff(fact_947_mult_Ocomm__neutral,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(A),one_one_complex) = A ).

% mult.comm_neutral
tff(fact_948_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_949_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_950_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_951_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_952_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_953_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_954_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_955_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_956_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_957_diff__diff__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) ).

% diff_diff_eq
tff(fact_958_diff__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),B2)),C) = aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ).

% diff_diff_eq
tff(fact_959_diff__diff__eq,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)) ).

% diff_diff_eq
tff(fact_960_diff__diff__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ).

% diff_diff_eq
tff(fact_961_add__implies__diff,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2) = A )
     => ( C = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) ) ) ).

% add_implies_diff
tff(fact_962_add__implies__diff,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2) = A )
     => ( C = aa_rat_rat(minus_minus_rat(A),B2) ) ) ).

% add_implies_diff
tff(fact_963_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2) ) ) ).

% add_implies_diff
tff(fact_964_add__implies__diff,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) ) ) ).

% add_implies_diff
tff(fact_965_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_966_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(A),C)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_967_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B2) ).

% diff_add_eq_diff_diff_swap
tff(fact_968_diff__add__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B2) ).

% diff_add_eq
tff(fact_969_diff__add__eq,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),B2)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B2) ).

% diff_add_eq
tff(fact_970_diff__add__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2) ).

% diff_add_eq
tff(fact_971_diff__diff__eq2,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B2) ).

% diff_diff_eq2
tff(fact_972_diff__diff__eq2,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(minus_minus_rat(B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B2) ).

% diff_diff_eq2
tff(fact_973_diff__diff__eq2,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2) ).

% diff_diff_eq2
tff(fact_974_add__diff__eq,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) ).

% add_diff_eq
tff(fact_975_add__diff__eq,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(minus_minus_rat(B2),C)) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) ).

% add_diff_eq
tff(fact_976_add__diff__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% add_diff_eq
tff(fact_977_eq__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = C ) ) ).

% eq_diff_eq
tff(fact_978_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( A = aa_rat_rat(minus_minus_rat(C),B2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = C ) ) ).

% eq_diff_eq
tff(fact_979_eq__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = C ) ) ).

% eq_diff_eq
tff(fact_980_diff__eq__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = C )
    <=> ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2) ) ) ).

% diff_eq_eq
tff(fact_981_diff__eq__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( aa_rat_rat(minus_minus_rat(A),B2) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2) ) ) ).

% diff_eq_eq
tff(fact_982_diff__eq__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2) ) ) ).

% diff_eq_eq
tff(fact_983_group__cancel_Osub1,axiom,
    ! [A3: real,K: real,A: real,B2: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A3),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_984_group__cancel_Osub1,axiom,
    ! [A3: rat,K: rat,A: rat,B2: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(minus_minus_rat(A3),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(minus_minus_rat(A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_985_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B2: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A3),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub1
tff(fact_986_add__less__le__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_987_add__less__le__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_988_add__less__le__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_989_add__less__le__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_less_le_mono
tff(fact_990_add__le__less__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_991_add__le__less__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_992_add__le__less__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_993_add__le__less__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),D)) ) ) ).

% add_le_less_mono
tff(fact_994_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_995_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_996_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_997_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_998_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K: real,L: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),J)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,K),L) )
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,I),K)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_999_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),J)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,K),L) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1000_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),L) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1001_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,K),L) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1002_diff__le__eq,axiom,
    ! [A: real,B2: 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),B2)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% diff_le_eq
tff(fact_1003_diff__le__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B2)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2)) ) ).

% diff_le_eq
tff(fact_1004_diff__le__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% diff_le_eq
tff(fact_1005_le__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) ) ).

% le_diff_eq
tff(fact_1006_le__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(minus_minus_rat(C),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) ) ).

% le_diff_eq
tff(fact_1007_le__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ) ).

% le_diff_eq
tff(fact_1008_diff__add,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),A) = B2 ) ) ).

% diff_add
tff(fact_1009_le__add__diff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => 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,B2),C)),A)) ) ).

% le_add_diff
tff(fact_1010_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),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)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1011_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B2)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1015_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),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)),B2) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1016_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A)) = B2 ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1017_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A) = C )
        <=> ( B2 = 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_1018_diff__less__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),C)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),B2)) ) ).

% diff_less_eq
tff(fact_1019_diff__less__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B2)),C)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B2)) ) ).

% diff_less_eq
tff(fact_1020_diff__less__eq,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B2)) ) ).

% diff_less_eq
tff(fact_1021_less__diff__eq,axiom,
    ! [A: real,C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,C),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),C) ) ).

% less_diff_eq
tff(fact_1022_less__diff__eq,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(minus_minus_rat(C),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) ) ).

% less_diff_eq
tff(fact_1023_less__diff__eq,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ) ).

% less_diff_eq
tff(fact_1024_real__average__minus__second,axiom,
    ! [B2: 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,B2),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,B2),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_second
tff(fact_1025_real__average__minus__first,axiom,
    ! [A: real,B2: 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),B2)),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,B2),A)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% real_average_minus_first
tff(fact_1026_times__divide__eq__left,axiom,
    ! [B2: complex,C: complex,A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(B2),C)),A) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(B2),A)),C) ).

% times_divide_eq_left
tff(fact_1027_times__divide__eq__left,axiom,
    ! [B2: real,C: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A)),C) ).

% times_divide_eq_left
tff(fact_1028_times__divide__eq__left,axiom,
    ! [B2: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(B2),C)),A) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),C) ).

% times_divide_eq_left
tff(fact_1029_divide__divide__eq__left,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B2)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(B2),C)) ).

% divide_divide_eq_left
tff(fact_1030_divide__divide__eq__left,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C)) ).

% divide_divide_eq_left
tff(fact_1031_divide__divide__eq__left,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B2)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ).

% divide_divide_eq_left
tff(fact_1032_divide__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(divide1717551699836669952omplex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),B2) ).

% divide_divide_eq_right
tff(fact_1033_divide__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B2) ).

% divide_divide_eq_right
tff(fact_1034_divide__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(divide_divide_rat(B2),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2) ).

% divide_divide_eq_right
tff(fact_1035_times__divide__eq__right,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),C) ).

% times_divide_eq_right
tff(fact_1036_times__divide__eq__right,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C) ).

% times_divide_eq_right
tff(fact_1037_times__divide__eq__right,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(divide_divide_rat(B2),C)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C) ).

% times_divide_eq_right
tff(fact_1038_less__half__sum,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))) ) ).

% less_half_sum
tff(fact_1039_less__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_1040_gt__half__sum,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => 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),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),one_one_real))),B2) ) ).

% gt_half_sum
tff(fact_1041_gt__half__sum,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B2) ) ).

% gt_half_sum
tff(fact_1042_discrete,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> 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)),B2) ) ).

% discrete
tff(fact_1043_discrete,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B2) ) ).

% discrete
tff(fact_1044_low__def,axiom,
    ! [Xb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% low_def
tff(fact_1045_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit0
tff(fact_1046_mod__mod__trivial,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mod_trivial
tff(fact_1047_mod__mod__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_mod_trivial
tff(fact_1048_mod__mod__trivial,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(modulo364778990260209775nteger(A,B2),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_mod_trivial
tff(fact_1049_mod__add__self2,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_add_self2
tff(fact_1050_mod__add__self2,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_add_self2
tff(fact_1051_mod__add__self2,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B2),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_add_self2
tff(fact_1052_mod__add__self1,axiom,
    ! [B2: 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,B2),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_add_self1
tff(fact_1053_mod__add__self1,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_add_self1
tff(fact_1054_mod__add__self1,axiom,
    ! [B2: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(B2,A),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_add_self1
tff(fact_1055_minus__mod__self2,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% minus_mod_self2
tff(fact_1056_minus__mod__self2,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B2),B2) = modulo364778990260209775nteger(A,B2) ).

% minus_mod_self2
tff(fact_1057_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_1058_mod__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = Mb ) ) ).

% mod_less
tff(fact_1059_mod__mult__self4,axiom,
    ! [B2: 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,B2),C)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self4
tff(fact_1060_mod__mult__self4,axiom,
    ! [B2: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_mult_self4
tff(fact_1061_mod__mult__self4,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C),A),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_mult_self4
tff(fact_1062_mod__mult__self3,axiom,
    ! [C: nat,B2: 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),B2)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self3
tff(fact_1063_mod__mult__self3,axiom,
    ! [C: int,B2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_mult_self3
tff(fact_1064_mod__mult__self3,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2),A),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_mult_self3
tff(fact_1065_mod__mult__self2,axiom,
    ! [A: nat,B2: 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,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self2
tff(fact_1066_mod__mult__self2,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_mult_self2
tff(fact_1067_mod__mult__self2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_mult_self2
tff(fact_1068_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B2: 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),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% mod_mult_self1
tff(fact_1069_mod__mult__self1,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% mod_mult_self1
tff(fact_1070_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)),B2) = modulo364778990260209775nteger(A,B2) ).

% mod_mult_self1
tff(fact_1071_Suc__mod__mult__self1,axiom,
    ! [Mb: 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,Mb),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,Mb)),Nb) ).

% Suc_mod_mult_self1
tff(fact_1072_Suc__mod__mult__self2,axiom,
    ! [Mb: 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,Mb),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,Mb)),Nb) ).

% Suc_mod_mult_self2
tff(fact_1073_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self3
tff(fact_1074_Suc__mod__mult__self4,axiom,
    ! [Nb: nat,K: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K)),Mb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% Suc_mod_mult_self4
tff(fact_1075_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_1076_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_1077_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_1078_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_1079_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_1080_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_1081_mod2__Suc__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% mod2_Suc_Suc
tff(fact_1082_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_1083_mod__mult__right__eq,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),C) ).

% mod_mult_right_eq
tff(fact_1084_mod__mult__right__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ).

% mod_mult_right_eq
tff(fact_1085_mod__mult__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) ).

% mod_mult_right_eq
tff(fact_1086_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B2: 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)),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) ).

% mod_mult_left_eq
tff(fact_1087_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ).

% mod_mult_left_eq
tff(fact_1088_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B2),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) ).

% mod_mult_left_eq
tff(fact_1089_mult__mod__right,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = 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),B2)) ).

% mult_mod_right
tff(fact_1090_mult__mod__right,axiom,
    ! [C: int,A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ).

% mult_mod_right
tff(fact_1091_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B2)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)) ).

% mult_mod_right
tff(fact_1092_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),C) ).

% mod_mult_mult2
tff(fact_1093_mod__mult__mult2,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),C) ).

% mod_mult_mult2
tff(fact_1094_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B2)),C) ).

% mod_mult_mult2
tff(fact_1095_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),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,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A2),B)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1096_mod__mult__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A2),B)),C) ) ) ) ).

% mod_mult_cong
tff(fact_1097_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A2,C) )
     => ( ( modulo364778990260209775nteger(B2,C) = modulo364778990260209775nteger(B,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A2),B),C) ) ) ) ).

% mod_mult_cong
tff(fact_1098_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2)),C) ).

% mod_mult_eq
tff(fact_1099_mod__mult__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ).

% mod_mult_eq
tff(fact_1100_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) ).

% mod_mult_eq
tff(fact_1101_mod__add__right__eq,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),C) ).

% mod_add_right_eq
tff(fact_1102_mod__add__right__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_right_eq
tff(fact_1103_mod__add__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B2),C) ).

% mod_add_right_eq
tff(fact_1104_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B2: 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)),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),C) ).

% mod_add_left_eq
tff(fact_1105_mod__add__left__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_left_eq
tff(fact_1106_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),B2),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B2),C) ).

% mod_add_left_eq
tff(fact_1107_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B2: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A2),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),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,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B)),C) ) ) ) ).

% mod_add_cong
tff(fact_1108_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B)),C) ) ) ) ).

% mod_add_cong
tff(fact_1109_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A2,C) )
     => ( ( modulo364778990260209775nteger(B2,C) = modulo364778990260209775nteger(B,C) )
       => ( modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B2),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A2,B),C) ) ) ) ).

% mod_add_cong
tff(fact_1110_mod__add__eq,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2)),C) ).

% mod_add_eq
tff(fact_1111_mod__add__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) ).

% mod_add_eq
tff(fact_1112_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(plus_p5714425477246183910nteger(A,B2),C) ).

% mod_add_eq
tff(fact_1113_mod__diff__right__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) ).

% mod_diff_right_eq
tff(fact_1114_mod__diff__right__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(A,modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B2),C) ).

% mod_diff_right_eq
tff(fact_1115_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) ).

% mod_diff_left_eq
tff(fact_1116_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),B2),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B2),C) ).

% mod_diff_left_eq
tff(fact_1117_mod__diff__cong,axiom,
    ! [A: int,C: int,A2: int,B2: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B)),C) ) ) ) ).

% mod_diff_cong
tff(fact_1118_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B2: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A2,C) )
     => ( ( modulo364778990260209775nteger(B2,C) = modulo364778990260209775nteger(B,C) )
       => ( modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B2),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A2,B),C) ) ) ) ).

% mod_diff_cong
tff(fact_1119_mod__diff__eq,axiom,
    ! [A: int,C: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) ).

% mod_diff_eq
tff(fact_1120_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B2,C)),C) = modulo364778990260209775nteger(minus_8373710615458151222nteger(A,B2),C) ).

% mod_diff_eq
tff(fact_1121_power__mod,axiom,
    ! [A: nat,B2: 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),B2)),Nb)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B2) ).

% power_mod
tff(fact_1122_power__mod,axiom,
    ! [A: int,B2: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),Nb)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(A),Nb)),B2) ).

% power_mod
tff(fact_1123_power__mod,axiom,
    ! [A: code_integer,B2: code_integer,Nb: nat] : modulo364778990260209775nteger(power_8256067586552552935nteger(modulo364778990260209775nteger(A,B2),Nb),B2) = modulo364778990260209775nteger(power_8256067586552552935nteger(A,Nb),B2) ).

% power_mod
tff(fact_1124_mod__Suc__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,Mb))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_1125_mod__Suc__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) ).

% mod_Suc_eq
tff(fact_1126_mod__less__eq__dividend,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Mb) ).

% mod_less_eq_dividend
tff(fact_1127_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_1128_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_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_1129_cong__exp__iff__simps_I9_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_1130_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,one)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Nb)),aa_num_nat(numeral_numeral_nat,one)) ).

% cong_exp_iff_simps(4)
tff(fact_1131_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,one)) = aa_int_int(aa_int_fun_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_1132_cong__exp__iff__simps_I4_J,axiom,
    ! [Mb: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(one)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) ).

% cong_exp_iff_simps(4)
tff(fact_1133_mod__geq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_1134_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
    <=> ? [Q1: nat,Q22: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q1)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1135_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_1136_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(6)
tff(fact_1137_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_1138_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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_1139_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1140_cong__exp__iff__simps_I8_J,axiom,
    ! [Mb: num,Q2: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) != modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) ).

% cong_exp_iff_simps(8)
tff(fact_1141_mod__eqE,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
     => ~ ! [D2: int] : B2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ).

% mod_eqE
tff(fact_1142_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B2,C) )
     => ~ ! [D2: code_integer] : B2 != plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ).

% mod_eqE
tff(fact_1143_div__add1__eq,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),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,B2),C))),C)) ).

% div_add1_eq
tff(fact_1144_div__add1__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C)) ).

% div_add1_eq
tff(fact_1145_div__add1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B2),C) = plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B2,C)),divide6298287555418463151nteger(plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,C),modulo364778990260209775nteger(B2,C)),C)) ).

% div_add1_eq
tff(fact_1146_mod__induct,axiom,
    ! [P: fun_nat_o,Nb: nat,P2: nat,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),P2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),P2)
         => ( ! [N2: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N2),P2)
               => ( aa_nat_o(P,N2)
                 => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,N2)),P2)) ) )
           => aa_nat_o(P,Mb) ) ) ) ) ).

% mod_induct
tff(fact_1147_mod__Suc__le__divisor,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_1148_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Y),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1149_mod__eq__nat2E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),Nb)
       => ~ ! [S3: nat] : Nb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_1150_mod__eq__nat1E,axiom,
    ! [Mb: nat,Q2: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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),Mb)
       => ~ ! [S3: nat] : Mb != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Q2),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_1151_mod__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb),Mb,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ).

% mod_if
tff(fact_1152_le__mod__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_1153_mult__div__mod__eq,axiom,
    ! [B2: 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) = A ).

% mult_div_mod_eq
tff(fact_1154_mult__div__mod__eq,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) = A ).

% mult_div_mod_eq
tff(fact_1155_mult__div__mod__eq,axiom,
    ! [B2: code_integer,A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(A,B2)),modulo364778990260209775nteger(A,B2)) = A ).

% mult_div_mod_eq
tff(fact_1156_mod__mult__div__eq,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))) = A ).

% mod_mult_div_eq
tff(fact_1157_mod__mult__div__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))) = A ).

% mod_mult_div_eq
tff(fact_1158_mod__mult__div__eq,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(A,B2))) = A ).

% mod_mult_div_eq
tff(fact_1159_mod__div__mult__eq,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),B2)) = A ).

% mod_div_mult_eq
tff(fact_1160_mod__div__mult__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2)) = A ).

% mod_div_mult_eq
tff(fact_1161_mod__div__mult__eq,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(modulo364778990260209775nteger(A,B2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2)) = A ).

% mod_div_mult_eq
tff(fact_1162_div__mult__mod__eq,axiom,
    ! [A: nat,B2: 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),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) = A ).

% div_mult_mod_eq
tff(fact_1163_div__mult__mod__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) = A ).

% div_mult_mod_eq
tff(fact_1164_div__mult__mod__eq,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2),modulo364778990260209775nteger(A,B2)) = A ).

% div_mult_mod_eq
tff(fact_1165_mod__div__decomp,axiom,
    ! [A: nat,B2: 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),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) ).

% mod_div_decomp
tff(fact_1166_mod__div__decomp,axiom,
    ! [A: int,B2: int] : A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ).

% mod_div_decomp
tff(fact_1167_mod__div__decomp,axiom,
    ! [A: code_integer,B2: code_integer] : A = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2),modulo364778990260209775nteger(A,B2)) ).

% mod_div_decomp
tff(fact_1168_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B2: 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),B2)),B2)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1169_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_1170_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2),modulo364778990260209775nteger(A,B2)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(1)
tff(fact_1171_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1172_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_1173_cancel__div__mod__rules_I2_J,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] : plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(A,B2)),modulo364778990260209775nteger(A,B2)),C) = plus_p5714425477246183910nteger(A,C) ).

% cancel_div_mod_rules(2)
tff(fact_1174_div__mult1__eq,axiom,
    ! [A: nat,B2: 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),B2)),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,B2),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,B2),C))),C)) ).

% div_mult1_eq
tff(fact_1175_div__mult1__eq,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C))),C)) ).

% div_mult1_eq
tff(fact_1176_div__mult1__eq,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B2,C)),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B2,C)),C)) ).

% div_mult1_eq
tff(fact_1177_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% minus_mult_div_eq_mod
tff(fact_1178_minus__mult__div__eq__mod,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% minus_mult_div_eq_mod
tff(fact_1179_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(A,B2))) = modulo364778990260209775nteger(A,B2) ).

% minus_mult_div_eq_mod
tff(fact_1180_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ).

% minus_mod_eq_mult_div
tff(fact_1181_minus__mod__eq__mult__div,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ).

% minus_mod_eq_mult_div
tff(fact_1182_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(A,B2)) ).

% minus_mod_eq_mult_div
tff(fact_1183_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_1184_minus__mod__eq__div__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_1185_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2) ).

% minus_mod_eq_div_mult
tff(fact_1186_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B2: 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),B2)),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ).

% minus_div_mult_eq_mod
tff(fact_1187_minus__div__mult__eq__mod,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ).

% minus_div_mult_eq_mod
tff(fact_1188_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),B2)) = modulo364778990260209775nteger(A,B2) ).

% minus_div_mult_eq_mod
tff(fact_1189_mod__mult2__eq,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),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,Mb),Nb)),Q2))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% mod_mult2_eq
tff(fact_1190_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : A3 = 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,A3),Nb)),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),Nb)) ).

% div_mod_decomp
tff(fact_1191_modulo__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_1192_linordered__field__no__lb,axiom,
    ! [X4: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),X4) ).

% linordered_field_no_lb
tff(fact_1193_linordered__field__no__lb,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),X4) ).

% linordered_field_no_lb
tff(fact_1194_linordered__field__no__ub,axiom,
    ! [X4: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,X4),X_1) ).

% linordered_field_no_ub
tff(fact_1195_linordered__field__no__ub,axiom,
    ! [X4: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),X_1) ).

% linordered_field_no_ub
tff(fact_1196_bounded__Max__nat,axiom,
    ! [P: fun_nat_o,Xb: nat,M5: nat] :
      ( aa_nat_o(P,Xb)
     => ( ! [X3: nat] :
            ( aa_nat_o(P,X3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),M5) )
       => ~ ! [M3: nat] :
              ( aa_nat_o(P,M3)
             => ~ ! [X4: nat] :
                    ( aa_nat_o(P,X4)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1197_div__exp__mod__exp__eq,axiom,
    ! [A: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1198_div__exp__mod__exp__eq,axiom,
    ! [A: int,Nb: nat,Mb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1199_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,Nb: nat,Mb: nat] : modulo364778990260209775nteger(divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb)) = divide6298287555418463151nteger(modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% div_exp_mod_exp_eq
tff(fact_1200_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1201_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1202_mult__exp__mod__exp__eq,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1203_divide__divide__eq__left_H,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(A),B2)),C) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(C),B2)) ).

% divide_divide_eq_left'
tff(fact_1204_divide__divide__eq__left_H,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2)) ).

% divide_divide_eq_left'
tff(fact_1205_divide__divide__eq__left_H,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(A),B2)),C) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ).

% divide_divide_eq_left'
tff(fact_1206_divide__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Zb: complex,W: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Zb),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),W)),aa_complex_complex(times_times_complex(Y),Zb)) ).

% divide_divide_times_eq
tff(fact_1207_divide__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Zb: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Zb),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),W)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb)) ).

% divide_divide_times_eq
tff(fact_1208_divide__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Zb: rat,W: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Zb),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),W)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb)) ).

% divide_divide_times_eq
tff(fact_1209_times__divide__times__eq,axiom,
    ! [Xb: complex,Y: complex,Zb: complex,W: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(Zb),W)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(Xb),Zb)),aa_complex_complex(times_times_complex(Y),W)) ).

% times_divide_times_eq
tff(fact_1210_times__divide__times__eq,axiom,
    ! [Xb: real,Y: real,Zb: real,W: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Zb),W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),W)) ).

% times_divide_times_eq
tff(fact_1211_times__divide__times__eq,axiom,
    ! [Xb: rat,Y: rat,Zb: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(Zb),W)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ).

% times_divide_times_eq
tff(fact_1212_add__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),B2)),C) = aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B2),C)) ).

% add_divide_distrib
tff(fact_1213_add__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),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,B2),C)) ).

% add_divide_distrib
tff(fact_1214_add__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B2),C)) ).

% add_divide_distrib
tff(fact_1215_diff__divide__distrib,axiom,
    ! [A: complex,B2: complex,C: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),B2)),C) = aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),C)),aa_complex_complex(divide1717551699836669952omplex(B2),C)) ).

% diff_divide_distrib
tff(fact_1216_diff__divide__distrib,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),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,B2),C)) ).

% diff_divide_distrib
tff(fact_1217_diff__divide__distrib,axiom,
    ! [A: rat,B2: rat,C: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),B2)),C) = aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),C)),aa_rat_rat(divide_divide_rat(B2),C)) ).

% diff_divide_distrib
tff(fact_1218_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_1219_unset__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% unset_bit_Suc
tff(fact_1220_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_1221_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_1222_flip__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% flip_bit_Suc
tff(fact_1223_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_1224_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_1225_set__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% set_bit_Suc
tff(fact_1226_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_1227_dbl__simps_I3_J,axiom,
    aa_complex_complex(neg_nu7009210354673126013omplex,one_one_complex) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% dbl_simps(3)
tff(fact_1228_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_1229_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_1230_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_1231_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B2: 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),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))
         => ( 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))),B2))),one_one_Code_integer) = divide6298287555418463151nteger(A,B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1232_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B2: 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),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),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))),B2)))
         => ( 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))),B2)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1233_divmod__digit__1_I1_J,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_1234_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_1235_signed__take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% signed_take_bit_Suc
tff(fact_1236_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(pow(K),L)) ).

% power_numeral
tff(fact_1237_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(pow(K),L)) ).

% power_numeral
tff(fact_1238_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(pow(K),L)) ).

% power_numeral
tff(fact_1239_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(pow(K),L)) ).

% power_numeral
tff(fact_1240_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(pow(K),L)) ).

% power_numeral
tff(fact_1241_arith__geo__mean,axiom,
    ! [U: real,Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,U),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1242_arith__geo__mean,axiom,
    ! [U: rat,Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ) ) ) ).

% arith_geo_mean
tff(fact_1243_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_1244_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_1245_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_1246_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_1247_le0,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Nb) ).

% le0
tff(fact_1248_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_1249_Nat_Oadd__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),zero_zero_nat) = Mb ).

% Nat.add_0_right
tff(fact_1250_add__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        & ( Nb = zero_zero_nat ) ) ) ).

% add_is_0
tff(fact_1251_diff__self__eq__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Mb) = zero_zero_nat ).

% diff_self_eq_0
tff(fact_1252_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_1253_mult__cancel2,axiom,
    ! [Mb: nat,K: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),K) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),K) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
tff(fact_1254_mult__cancel1,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( Mb = Nb )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
tff(fact_1255_mult__0__right,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),zero_zero_nat) = zero_zero_nat ).

% mult_0_right
tff(fact_1256_mult__is__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = zero_zero_nat )
    <=> ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ) ) ).

% mult_is_0
tff(fact_1257_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
tff(fact_1258_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
tff(fact_1259_i0__less,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
    <=> ( Nb != zero_z5237406670263579293d_enat ) ) ).

% i0_less
tff(fact_1260_idiff__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(zero_z5237406670263579293d_enat,Nb) = zero_z5237406670263579293d_enat ).

% idiff_0
tff(fact_1261_idiff__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,zero_z5237406670263579293d_enat) = Nb ).

% idiff_0_right
tff(fact_1262_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_1263_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_1264_mult__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1265_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1266_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1267_mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_right
tff(fact_1268_mult__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1269_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1270_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1271_mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( A = B2 ) ) ) ).

% mult_cancel_left
tff(fact_1272_mult__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% mult_eq_0_iff
tff(fact_1273_mult__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_1274_mult__eq__0__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_1275_mult__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_1276_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_1277_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_1278_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_1279_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_1280_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_1281_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_1282_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_1283_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_1284_add__0,axiom,
    ! [A: literal] : plus_plus_literal(zero_zero_literal,A) = A ).

% add_0
tff(fact_1285_add__0,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,zero_zero_real),A) = A ).

% add_0
tff(fact_1286_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_1287_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_1288_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_1289_zero__eq__add__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1290_add__eq__0__iff__both__eq__0,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
    <=> ( ( Xb = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1291_add__cancel__right__right,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_right
tff(fact_1292_add__cancel__right__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_1293_add__cancel__right__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_1294_add__cancel__right__right,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_1295_add__cancel__right__left,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_right_left
tff(fact_1296_add__cancel__right__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_1297_add__cancel__right__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_1298_add__cancel__right__left,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_1299_add__cancel__left__right,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_right
tff(fact_1300_add__cancel__left__right,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = A )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_1301_add__cancel__left__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_1302_add__cancel__left__right,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_1303_add__cancel__left__left,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) = A )
    <=> ( B2 = zero_zero_real ) ) ).

% add_cancel_left_left
tff(fact_1304_add__cancel__left__left,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) = A )
    <=> ( B2 = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_1305_add__cancel__left__left,axiom,
    ! [B2: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) = A )
    <=> ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_1306_add__cancel__left__left,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) = A )
    <=> ( B2 = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_1307_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_1308_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_1309_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_1310_add_Oright__neutral,axiom,
    ! [A: literal] : plus_plus_literal(A,zero_zero_literal) = A ).

% add.right_neutral
tff(fact_1311_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_1312_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_1313_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_1314_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_1315_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_1316_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_1317_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_1318_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1319_diff__zero,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),zero_zero_real) = A ).

% diff_zero
tff(fact_1320_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_zero
tff(fact_1321_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_1322_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_1323_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_1324_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_1325_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_1326_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_1327_diff__self,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),A) = zero_zero_real ).

% diff_self
tff(fact_1328_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(minus_minus_rat(A),A) = zero_zero_rat ).

% diff_self
tff(fact_1329_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_1330_div__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(zero_zero_complex),A) = zero_zero_complex ).

% div_0
tff(fact_1331_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_1332_div__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(zero_zero_rat),A) = zero_zero_rat ).

% div_0
tff(fact_1333_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_1334_div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% div_0
tff(fact_1335_div__by__0,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% div_by_0
tff(fact_1336_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_1337_div__by__0,axiom,
    ! [A: rat] : aa_rat_rat(divide_divide_rat(A),zero_zero_rat) = zero_zero_rat ).

% div_by_0
tff(fact_1338_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_1339_div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% div_by_0
tff(fact_1340_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_1341_bits__div__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,zero_zero_int),A) = zero_zero_int ).

% bits_div_0
tff(fact_1342_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_1343_bits__div__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),zero_zero_int) = zero_zero_int ).

% bits_div_by_0
tff(fact_1344_divide__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
    <=> ( ( A = zero_zero_complex )
        | ( B2 = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
tff(fact_1345_divide__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = zero_zero_real )
    <=> ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divide_eq_0_iff
tff(fact_1346_divide__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B2) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
tff(fact_1347_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(C),A) = aa_complex_complex(divide1717551699836669952omplex(C),B2) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1348_divide__cancel__left,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1349_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(C),A) = aa_rat_rat(divide_divide_rat(C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_left
tff(fact_1350_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> ( ( C = zero_zero_complex )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1351_divide__cancel__right,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1352_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B2 ) ) ) ).

% divide_cancel_right
tff(fact_1353_division__ring__divide__zero,axiom,
    ! [A: complex] : aa_complex_complex(divide1717551699836669952omplex(A),zero_zero_complex) = zero_zero_complex ).

% division_ring_divide_zero
tff(fact_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_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_1363_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_1364_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_1365_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_1366_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B2,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_1367_dvd__add__triv__right__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_1368_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_1369_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_1370_dvd__add__triv__right__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ).

% dvd_add_triv_right_iff
tff(fact_1371_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(A,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_1372_dvd__add__triv__left__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_1373_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_1374_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B2: 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),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_1375_dvd__add__triv__left__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ).

% dvd_add_triv_left_iff
tff(fact_1376_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_1377_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_1378_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_1379_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_1380_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_1381_mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% mod_0
tff(fact_1382_mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% mod_0
tff(fact_1383_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_1384_mod__by__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),zero_zero_int) = A ).

% mod_by_0
tff(fact_1385_mod__by__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,zero_z3403309356797280102nteger) = A ).

% mod_by_0
tff(fact_1386_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_1387_mod__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),A) = zero_zero_int ).

% mod_self
tff(fact_1388_mod__self,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,A) = zero_z3403309356797280102nteger ).

% mod_self
tff(fact_1389_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_1390_bits__mod__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,zero_zero_int),A) = zero_zero_int ).

% bits_mod_0
tff(fact_1391_bits__mod__0,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(zero_z3403309356797280102nteger,A) = zero_z3403309356797280102nteger ).

% bits_mod_0
tff(fact_1392_div__dvd__div,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( 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(B2,A)),divide6298287555418463151nteger(C,A))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_1393_div__dvd__div,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( 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,B2),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,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_1394_div__dvd__div,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( 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(aa_int_fun_int_int(divide_divide_int,B2),A)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ) ).

% div_dvd_div
tff(fact_1395_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_1396_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_1397_add__gr__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% add_gr_0
tff(fact_1398_one__eq__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% one_eq_mult_iff
tff(fact_1399_mult__eq__1__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% mult_eq_1_iff
tff(fact_1400_div__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = Mb ).

% div_by_Suc_0
tff(fact_1401_zero__less__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% zero_less_diff
tff(fact_1402_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1403_nat__0__less__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_1404_mult__less__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ) ) ).

% mult_less_cancel2
tff(fact_1405_div__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat ) ) ).

% div_less
tff(fact_1406_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb))
    <=> ( Xb = zero_zero_real ) ) ).

% not_real_square_gt_zero
tff(fact_1407_diff__is__0__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% diff_is_0_eq
tff(fact_1408_diff__is__0__eq_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat ) ) ).

% diff_is_0_eq'
tff(fact_1409_dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(suc,zero_zero_nat))
    <=> ( Mb = aa_nat_nat(suc,zero_zero_nat) ) ) ).

% dvd_1_iff_1
tff(fact_1410_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_1411_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_1412_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_1413_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = zero_zero_nat )
        | ( Xb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1414_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1415_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_1416_mod__by__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% mod_by_Suc_0
tff(fact_1417_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_1418_nat__dvd__1__iff__1,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),one_one_nat)
    <=> ( Mb = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_1419_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_1420_dbl__simps_I2_J,axiom,
    aa_real_real(neg_numeral_dbl_real,zero_zero_real) = zero_zero_real ).

% dbl_simps(2)
tff(fact_1421_dbl__simps_I2_J,axiom,
    aa_rat_rat(neg_numeral_dbl_rat,zero_zero_rat) = zero_zero_rat ).

% dbl_simps(2)
tff(fact_1422_dbl__simps_I2_J,axiom,
    aa_int_int(neg_numeral_dbl_int,zero_zero_int) = zero_zero_int ).

% dbl_simps(2)
tff(fact_1423_add__le__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel1
tff(fact_1424_add__le__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_1425_add__le__same__cancel1,axiom,
    ! [B2: 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,B2),A)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_1426_add__le__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_1427_add__le__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% add_le_same_cancel2
tff(fact_1428_add__le__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_1429_add__le__same__cancel2,axiom,
    ! [A: nat,B2: 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),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_1430_add__le__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_1431_le__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel1
tff(fact_1432_le__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel1
tff(fact_1433_le__add__same__cancel1,axiom,
    ! [A: nat,B2: 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),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel1
tff(fact_1434_le__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel1
tff(fact_1435_le__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2) ) ).

% le_add_same_cancel2
tff(fact_1436_le__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2) ) ).

% le_add_same_cancel2
tff(fact_1437_le__add__same__cancel2,axiom,
    ! [A: nat,B2: 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,B2),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2) ) ).

% le_add_same_cancel2
tff(fact_1438_le__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2) ) ).

% le_add_same_cancel2
tff(fact_1439_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_1440_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1442_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_1443_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1444_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1445_diff__ge__0__iff__ge,axiom,
    ! [A: real,B2: 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),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1446_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1447_diff__ge__0__iff__ge,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% diff_ge_0_iff_ge
tff(fact_1448_add__less__same__cancel1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel1
tff(fact_1449_add__less__same__cancel1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_1450_add__less__same__cancel1,axiom,
    ! [B2: 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,B2),A)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_1451_add__less__same__cancel1,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_1452_add__less__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% add_less_same_cancel2
tff(fact_1453_add__less__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_1454_add__less__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),B2)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_1455_add__less__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_1456_less__add__same__cancel1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel1
tff(fact_1457_less__add__same__cancel1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel1
tff(fact_1458_less__add__same__cancel1,axiom,
    ! [A: nat,B2: 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),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel1
tff(fact_1459_less__add__same__cancel1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel1
tff(fact_1460_less__add__same__cancel2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2) ) ).

% less_add_same_cancel2
tff(fact_1461_less__add__same__cancel2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2) ) ).

% less_add_same_cancel2
tff(fact_1462_less__add__same__cancel2,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2) ) ).

% less_add_same_cancel2
tff(fact_1463_less__add__same__cancel2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ).

% less_add_same_cancel2
tff(fact_1464_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_1465_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1466_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1467_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_1468_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1469_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1470_diff__gt__0__iff__gt,axiom,
    ! [A: real,B2: 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),B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1471_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(minus_minus_rat(A),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1472_diff__gt__0__iff__gt,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% diff_gt_0_iff_gt
tff(fact_1473_mult__cancel__left1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C = aa_complex_complex(times_times_complex(C),B2) )
    <=> ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_left1
tff(fact_1474_mult__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_left1
tff(fact_1475_mult__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1476_mult__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1477_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_1478_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_1479_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1480_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1481_mult__cancel__right1,axiom,
    ! [C: complex,B2: complex] :
      ( ( C = aa_complex_complex(times_times_complex(B2),C) )
    <=> ( ( C = zero_zero_complex )
        | ( B2 = one_one_complex ) ) ) ).

% mult_cancel_right1
tff(fact_1482_mult__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( ( C = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C) )
    <=> ( ( C = zero_zero_real )
        | ( B2 = one_one_real ) ) ) ).

% mult_cancel_right1
tff(fact_1483_mult__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B2 = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1484_mult__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
    <=> ( ( C = zero_zero_int )
        | ( B2 = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1485_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_1486_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_1487_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1488_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1489_sum__squares__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1490_sum__squares__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1491_sum__squares__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1492_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ).

% div_mult_mult1_if
tff(fact_1493_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ).

% div_mult_mult1_if
tff(fact_1494_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ).

% div_mult_mult2
tff(fact_1495_div__mult__mult2,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ).

% div_mult_mult2
tff(fact_1496_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ).

% div_mult_mult1
tff(fact_1497_div__mult__mult1,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ).

% div_mult_mult1
tff(fact_1498_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1499_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1500_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1501_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B2: 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),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1502_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),A) = B2 ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1503_nonzero__mult__div__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1504_nonzero__mult__div__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1505_nonzero__mult__div__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1506_nonzero__mult__div__cancel__right,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1507_nonzero__mult__div__cancel__right,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1508_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B2: complex] :
      aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)) = $ite(C = zero_zero_complex,zero_zero_complex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1509_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B2: 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),B2)) = $ite(C = zero_zero_real,zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1510_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B2: rat] :
      aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(divide_divide_rat(A),B2)) ).

% mult_divide_mult_cancel_left_if
tff(fact_1511_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(C),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1512_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1513_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) = aa_rat_rat(divide_divide_rat(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1514_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(C),A)),aa_complex_complex(times_times_complex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1515_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B2: 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,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1516_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) = aa_rat_rat(divide_divide_rat(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1517_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(B2),C)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1518_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B2: 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,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1519_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) = aa_rat_rat(divide_divide_rat(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1520_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),C)),aa_complex_complex(times_times_complex(C),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1521_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B2: 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),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1522_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) = aa_rat_rat(divide_divide_rat(A),B2) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1523_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_1524_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_1525_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_1526_diff__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),one_one_int) = zero_zero_int ).

% diff_numeral_special(9)
tff(fact_1527_diff__add__zero,axiom,
    ! [A: nat,B2: 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),B2)) = zero_zero_nat ).

% diff_add_zero
tff(fact_1528_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% div_self
tff(fact_1529_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_1530_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% div_self
tff(fact_1531_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_1532_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_1533_divide__eq__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_1534_divide__eq__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = one_one_real )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_1535_divide__eq__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B2) = one_one_rat )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_1_iff
tff(fact_1536_one__eq__divide__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( one_one_complex = aa_complex_complex(divide1717551699836669952omplex(A),B2) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_1537_one__eq__divide__iff,axiom,
    ! [A: real,B2: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_1538_one__eq__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(A),B2) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% one_eq_divide_iff
tff(fact_1539_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),A) = one_one_complex ) ) ).

% divide_self
tff(fact_1540_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_1541_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_1542_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_1543_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_1544_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_1545_divide__eq__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A) = one_one_real )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_1546_divide__eq__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% divide_eq_eq_1
tff(fact_1547_eq__divide__eq__1,axiom,
    ! [B2: real,A: real] :
      ( ( one_one_real = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A) )
    <=> ( ( A != zero_zero_real )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_1548_eq__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(divide_divide_rat(B2),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B2 ) ) ) ).

% eq_divide_eq_1
tff(fact_1549_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_1550_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_1551_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_1552_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_1553_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B2: 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,B2),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1554_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B2: 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,B2),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,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1555_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_1556_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B2: 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),B2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1557_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B2: 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),B2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1558_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_1559_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B2: 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,B2),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_1560_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_1561_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_1562_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% dvd_mult_cancel_right
tff(fact_1563_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B2: 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),B2))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_1564_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> ( ( C = zero_zero_real )
        | aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_1565_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_1566_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% dvd_mult_cancel_left
tff(fact_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_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_1574_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_1575_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_1576_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_1577_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_1578_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_1579_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_1580_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_1581_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_1582_unit__prod,axiom,
    ! [A: code_integer,B2: 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,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_1583_unit__prod,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),one_one_nat) ) ) ).

% unit_prod
tff(fact_1584_unit__prod,axiom,
    ! [A: int,B2: 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,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int) ) ) ).

% unit_prod
tff(fact_1585_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1586_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B2: 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)),B2))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1587_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1588_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B2: 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)),B2))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1589_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ).

% dvd_add_times_triv_left_iff
tff(fact_1590_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1591_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),A)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1592_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1593_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B2: 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,B2),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),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1594_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ).

% dvd_add_times_triv_right_iff
tff(fact_1595_mod__mult__self1__is__0,axiom,
    ! [B2: 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,B2),A)),B2) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_1596_mod__mult__self1__is__0,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),B2) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_1597_mod__mult__self1__is__0,axiom,
    ! [B2: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_1598_mod__mult__self2__is__0,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),B2) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_1599_mod__mult__self2__is__0,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),B2) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_1600_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),B2) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_1601_dvd__div__mult__self,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B2,A)),A) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_1602_dvd__div__mult__self,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)),A) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_1603_dvd__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A)),A) = B2 ) ) ).

% dvd_div_mult_self
tff(fact_1604_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B2,A)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_1605_dvd__mult__div__cancel,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_1606_dvd__mult__div__cancel,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A)) = B2 ) ) ).

% dvd_mult_div_cancel
tff(fact_1607_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_1608_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_1609_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_1610_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_1611_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_1612_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_1613_unit__div,axiom,
    ! [A: code_integer,B2: 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,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B2)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_1614_unit__div,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),one_one_nat) ) ) ).

% unit_div
tff(fact_1615_unit__div,axiom,
    ! [A: int,B2: 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,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int) ) ) ).

% unit_div
tff(fact_1616_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_1617_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_1618_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(aa_int_fun_int_int(divide_divide_int,one_one_int),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_1619_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_1620_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_1621_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(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_1622_div__add,axiom,
    ! [C: code_integer,A: code_integer,B2: 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),B2)
       => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B2),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B2,C)) ) ) ) ).

% div_add
tff(fact_1623_div__add,axiom,
    ! [C: nat,A: nat,B2: 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),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),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,B2),C)) ) ) ) ).

% div_add
tff(fact_1624_div__add,axiom,
    ! [C: int,A: int,B2: 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),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ) ).

% div_add
tff(fact_1625_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B2: 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),B2)
       => ( divide6298287555418463151nteger(minus_8373710615458151222nteger(A,B2),C) = minus_8373710615458151222nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B2,C)) ) ) ) ).

% div_diff
tff(fact_1626_div__diff,axiom,
    ! [C: int,A: int,B2: 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),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ) ).

% div_diff
tff(fact_1627_mod__div__trivial,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) = zero_zero_nat ).

% mod_div_trivial
tff(fact_1628_mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) = zero_zero_int ).

% mod_div_trivial
tff(fact_1629_mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B2),B2) = zero_z3403309356797280102nteger ).

% mod_div_trivial
tff(fact_1630_bits__mod__div__trivial,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) = zero_zero_nat ).

% bits_mod_div_trivial
tff(fact_1631_bits__mod__div__trivial,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) = zero_zero_int ).

% bits_mod_div_trivial
tff(fact_1632_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B2: code_integer] : divide6298287555418463151nteger(modulo364778990260209775nteger(A,B2),B2) = zero_z3403309356797280102nteger ).

% bits_mod_div_trivial
tff(fact_1633_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W))) ).

% zmod_numeral_Bit0
tff(fact_1634_dvd__imp__mod__0,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_imp_mod_0
tff(fact_1635_dvd__imp__mod__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A) = zero_zero_int ) ) ).

% dvd_imp_mod_0
tff(fact_1636_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( modulo364778990260209775nteger(B2,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
tff(fact_1637_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_1638_one__le__mult__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ) ) ).

% one_le_mult_iff
tff(fact_1639_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1640_mult__le__cancel2,axiom,
    ! [Mb: 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,Mb),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,Mb),Nb) ) ) ).

% mult_le_cancel2
tff(fact_1641_div__mult__self1__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Nb) = Mb ) ) ).

% div_mult_self1_is_m
tff(fact_1642_div__mult__self__is__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Nb) = Mb ) ) ).

% div_mult_self_is_m
tff(fact_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_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_1654_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_1655_less__divide__eq__1__pos,axiom,
    ! [A: real,B2: 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,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_1656_less__divide__eq__1__pos,axiom,
    ! [A: rat,B2: 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(B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% less_divide_eq_1_pos
tff(fact_1657_less__divide__eq__1__neg,axiom,
    ! [A: real,B2: 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,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1658_less__divide__eq__1__neg,axiom,
    ! [A: rat,B2: 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(B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_1659_divide__less__eq__1__pos,axiom,
    ! [A: real,B2: 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,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1660_divide__less__eq__1__pos,axiom,
    ! [A: rat,B2: 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(B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_1661_divide__less__eq__1__neg,axiom,
    ! [A: real,B2: 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,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_1662_divide__less__eq__1__neg,axiom,
    ! [A: rat,B2: 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(B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% divide_less_eq_1_neg
tff(fact_1663_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_1664_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_1665_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W)) = A )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1666_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_num_real(numeral_numeral_real,W)) = A )
    <=> $ite(aa_num_real(numeral_numeral_real,W) != zero_zero_real,B2 = 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_1667_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) = A )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_1668_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),numera6690914467698888265omplex(W)) )
    <=> $ite(numera6690914467698888265omplex(W) != zero_zero_complex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W)) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1669_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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)) = B2,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1670_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B2),aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(aa_num_rat(numeral_numeral_rat,W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_num_rat(numeral_numeral_rat,W)) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_1671_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1672_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B2: 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),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1673_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),B2) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1674_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1675_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1676_nonzero__divide__mult__cancel__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1677_div__mult__self1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != 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),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self1
tff(fact_1678_div__mult__self1,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self1
tff(fact_1679_div__mult__self2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != 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,B2),C))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self2
tff(fact_1680_div__mult__self2,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self2
tff(fact_1681_div__mult__self3,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != 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),B2)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self3
tff(fact_1682_div__mult__self3,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self3
tff(fact_1683_div__mult__self4,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( ( B2 != 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,B2),C)),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) ) ) ).

% div_mult_self4
tff(fact_1684_div__mult__self4,axiom,
    ! [B2: int,C: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_mult_self4
tff(fact_1685_power__mono__iff,axiom,
    ! [A: real,B2: 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),B2)
       => ( 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,B2),Nb))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_1686_power__mono__iff,axiom,
    ! [A: rat,B2: 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),B2)
       => ( 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(B2),Nb))
          <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_1687_power__mono__iff,axiom,
    ! [A: nat,B2: 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),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(power_power_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
          <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_1688_power__mono__iff,axiom,
    ! [A: int,B2: 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),B2)
       => ( 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(B2),Nb))
          <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ) ) ).

% power_mono_iff
tff(fact_1689_unit__mult__div__div,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B2,A) ) ) ).

% unit_mult_div_div
tff(fact_1690_unit__mult__div__div,axiom,
    ! [A: nat,B2: 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,B2),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,B2),A) ) ) ).

% unit_mult_div_div
tff(fact_1691_unit__mult__div__div,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) ) ) ).

% unit_mult_div_div
tff(fact_1692_unit__div__mult__self,axiom,
    ! [A: code_integer,B2: 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(B2,A)),A) = B2 ) ) ).

% unit_div_mult_self
tff(fact_1693_unit__div__mult__self,axiom,
    ! [A: nat,B2: 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,B2),A)),A) = B2 ) ) ).

% unit_div_mult_self
tff(fact_1694_unit__div__mult__self,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A)),A) = B2 ) ) ).

% unit_div_mult_self
tff(fact_1695_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(aa_int_fun_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_1696_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_1697_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int) ) ).

% half_negative_int_iff
tff(fact_1698_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) ) ).

% half_nonnegative_int_iff
tff(fact_1699_le__divide__eq__1__pos,axiom,
    ! [A: real,B2: 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,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_1700_le__divide__eq__1__pos,axiom,
    ! [A: rat,B2: 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(B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% le_divide_eq_1_pos
tff(fact_1701_le__divide__eq__1__neg,axiom,
    ! [A: real,B2: 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,B2),A))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1702_le__divide__eq__1__neg,axiom,
    ! [A: rat,B2: 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(B2),A))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_1703_divide__le__eq__1__pos,axiom,
    ! [A: real,B2: 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,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1704_divide__le__eq__1__pos,axiom,
    ! [A: rat,B2: 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(B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_1705_divide__le__eq__1__neg,axiom,
    ! [A: real,B2: 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,B2),A)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_1706_divide__le__eq__1__neg,axiom,
    ! [A: rat,B2: 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(B2),A)),one_one_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% divide_le_eq_1_neg
tff(fact_1707_power__strict__decreasing__iff,axiom,
    ! [B2: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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,B2),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1708_power__strict__decreasing__iff,axiom,
    ! [B2: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(B2),Mb)),aa_nat_rat(power_power_rat(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1709_power__strict__decreasing__iff,axiom,
    ! [B2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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,B2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1710_power__strict__decreasing__iff,axiom,
    ! [B2: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),Mb)),aa_nat_int(power_power_int(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1711_even__mult__iff,axiom,
    ! [A: code_integer,B2: 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),B2))
    <=> ( 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))),B2) ) ) ).

% even_mult_iff
tff(fact_1712_even__mult__iff,axiom,
    ! [A: nat,B2: 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),B2))
    <=> ( 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))),B2) ) ) ).

% even_mult_iff
tff(fact_1713_even__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
    <=> ( 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))),B2) ) ) ).

% even_mult_iff
tff(fact_1714_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_1715_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_1716_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_1717_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_1718_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_1719_odd__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B2))
    <=> ~ ( ~ 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))),B2) ) ) ).

% odd_add
tff(fact_1720_odd__add,axiom,
    ! [A: nat,B2: 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),B2))
    <=> ~ ( ~ 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))),B2) ) ) ).

% odd_add
tff(fact_1721_odd__add,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> ~ ( ~ 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))),B2) ) ) ).

% odd_add
tff(fact_1722_even__add,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B2))
    <=> ( 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))),B2) ) ) ).

% even_add
tff(fact_1723_even__add,axiom,
    ! [A: nat,B2: 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),B2))
    <=> ( 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))),B2) ) ) ).

% even_add
tff(fact_1724_even__add,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))
    <=> ( 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))),B2) ) ) ).

% even_add
tff(fact_1725_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_1726_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(aa_int_fun_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_1727_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_1728_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_1729_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_1730_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_1731_add__self__mod__2,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Mb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% add_self_mod_2
tff(fact_1732_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_1733_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_1734_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_1735_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_1736_power__decreasing__iff,axiom,
    ! [B2: real,Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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,B2),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1737_power__decreasing__iff,axiom,
    ! [B2: rat,Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),one_one_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(B2),Mb)),aa_nat_rat(power_power_rat(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1738_power__decreasing__iff,axiom,
    ! [B2: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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,B2),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1739_power__decreasing__iff,axiom,
    ! [B2: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),Mb)),aa_nat_int(power_power_int(B2),Nb))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ) ) ) ).

% power_decreasing_iff
tff(fact_1740_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_1741_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_1742_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_1743_power2__eq__iff__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1744_power2__eq__iff__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1745_power2__eq__iff__nonneg,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1746_power2__eq__iff__nonneg,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
        <=> ( Xb = Y ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_1747_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_1748_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_1749_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_1750_sum__power2__eq__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_rat )
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1751_sum__power2__eq__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1752_sum__power2__eq__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_int )
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_1753_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_1754_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_1755_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_plus_one_iff
tff(fact_1756_even__diff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),minus_8373710615458151222nteger(A,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B2)) ) ).

% even_diff
tff(fact_1757_even__diff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ).

% even_diff
tff(fact_1758_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_1759_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_1760_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_1761_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_1762_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_1763_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_1764_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))),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_1765_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_1766_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_1767_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_1768_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_1769_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_1770_mod2__gr__0,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_1771_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% unset_bit_0
tff(fact_1772_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_1773_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_1774_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_1775_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_1776_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_1777_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_1778_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_two
tff(fact_1779_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_1780_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_1781_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% even_succ_div_2
tff(fact_1782_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(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_1783_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_1784_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1785_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_1786_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_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_even__diff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ) ) ).

% even_diff_nat
tff(fact_1798_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_1799_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_1800_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_1801_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_1802_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_1803_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_1804_set__bit__0,axiom,
    ! [A: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% set_bit_0
tff(fact_1805_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_1806_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),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1807_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_1808_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% even_succ_div_exp
tff(fact_1809_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_1810_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1811_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),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = plus_p5714425477246183910nteger(one_one_Code_integer,modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb))) ) ) ) ).

% even_succ_mod_exp
tff(fact_1812_split__zdiv,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_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)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_1813_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L)),one_one_int) ) ) ) ).

% div_pos_geq
tff(fact_1814_int__div__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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,B2),R2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1815_int__div__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2)
         => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1816_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: int] : A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)),Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),Nb)) ).

% div_mod_decomp_int
tff(fact_1817_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(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_neg_lemma
tff(fact_1818_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(aa_int_fun_int_int(divide_divide_int,Nb),K)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Nb),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
              & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
              & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
           => aa_int_o(aa_int_fun_int_o(P,I2),J3) ) ) ) ).

% split_pos_lemma
tff(fact_1819_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_1820_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,
              aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),Nb) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_1821_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ).

% zdiv_zmult2_eq
tff(fact_1822_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_1823_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B2: 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(aa_int_fun_int_int(divide_divide_int,A),B2))
      <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1824_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1825_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1826_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1827_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),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_1828_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),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_1829_div__nonpos__pos__le0,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),zero_zero_int) ) ) ).

% div_nonpos_pos_le0
tff(fact_1830_div__nonneg__neg__le0,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),zero_zero_int) ) ) ).

% div_nonneg_neg_le0
tff(fact_1831_div__neg__pos__less0,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),zero_zero_int) ) ) ).

% div_neg_pos_less0
tff(fact_1832_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L)) ) ) ).

% div_positive_int
tff(fact_1833_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L))
    <=> ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int) ) ) ) ).

% div_int_pos_iff
tff(fact_1834_zdiv__mono2__neg,axiom,
    ! [A: int,B: int,B2: 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_eq_int,B),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1835_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1836_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,I),K) = zero_zero_int )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1837_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A3),B3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),Nb) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B3),Nb) = zero_zero_int )
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B3),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_1838_zdiv__mono2,axiom,
    ! [A: int,B: int,B2: 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),B2)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ) ).

% zdiv_mono2
tff(fact_1839_zdiv__mono1,axiom,
    ! [A: int,A2: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A2),B2)) ) ) ).

% zdiv_mono1
tff(fact_1840_dvd__field__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
    <=> ( ( A = zero_zero_real )
       => ( B2 = zero_zero_real ) ) ) ).

% dvd_field_iff
tff(fact_1841_dvd__field__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
    <=> ( ( A = zero_zero_rat )
       => ( B2 = zero_zero_rat ) ) ) ).

% dvd_field_iff
tff(fact_1842_dvd__refl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),A) ).

% dvd_refl
tff(fact_1843_dvd__refl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),A) ).

% dvd_refl
tff(fact_1844_dvd__refl,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),A) ).

% dvd_refl
tff(fact_1845_dvd__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_trans
tff(fact_1846_dvd__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_trans
tff(fact_1847_dvd__trans,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_trans
tff(fact_1848_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_1849_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_1850_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_1851_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_1852_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_1853_dvd__antisym,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)
       => ( Mb = Nb ) ) ) ).

% dvd_antisym
tff(fact_1854_zero__reorient,axiom,
    ! [Xb: literal] :
      ( ( zero_zero_literal = Xb )
    <=> ( Xb = zero_zero_literal ) ) ).

% zero_reorient
tff(fact_1855_zero__reorient,axiom,
    ! [Xb: real] :
      ( ( zero_zero_real = Xb )
    <=> ( Xb = zero_zero_real ) ) ).

% zero_reorient
tff(fact_1856_zero__reorient,axiom,
    ! [Xb: rat] :
      ( ( zero_zero_rat = Xb )
    <=> ( Xb = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1857_zero__reorient,axiom,
    ! [Xb: nat] :
      ( ( zero_zero_nat = Xb )
    <=> ( Xb = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1858_zero__reorient,axiom,
    ! [Xb: int] :
      ( ( zero_zero_int = Xb )
    <=> ( Xb = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1859_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_1860_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_1861_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_1862_dvd__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ( ( divide6298287555418463151nteger(A,B2) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1863_dvd__div__eq__0__iff,axiom,
    ! [B2: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A)
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1864_dvd__div__eq__0__iff,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1865_dvd__div__eq__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
     => ( ( aa_rat_rat(divide_divide_rat(A),B2) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1866_dvd__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1867_dvd__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1868_nat__dvd__not__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ) ).

% nat_dvd_not_less
tff(fact_1869_mod__0__imp__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% mod_0_imp_dvd
tff(fact_1870_mod__0__imp__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% mod_0_imp_dvd
tff(fact_1871_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B2) = zero_z3403309356797280102nteger )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A) ) ).

% mod_0_imp_dvd
tff(fact_1872_dvd__eq__mod__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),A) = zero_zero_nat ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1873_dvd__eq__mod__eq__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A) = zero_zero_int ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1874_dvd__eq__mod__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
    <=> ( modulo364778990260209775nteger(B2,A) = zero_z3403309356797280102nteger ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1875_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat )
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1876_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1877_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B2) = zero_z3403309356797280102nteger )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A) ) ).

% mod_eq_0_iff_dvd
tff(fact_1878_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ~ ! [N2: nat] : Xb != aa_nat_nat(suc,N2) ) ).

% list_decode.cases
tff(fact_1879_pos__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_1880_neg__zmod__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_1881_unit__dvdE,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B2 != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_1882_unit__dvdE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B2 != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_1883_unit__dvdE,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B2 != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_1884_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
       => ( ( divide6298287555418463151nteger(B2,A) = C )
        <=> ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1885_dvd__div__eq__mult,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),A) = C )
        <=> ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1886_dvd__div__eq__mult,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) = C )
        <=> ( B2 = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_1887_div__dvd__iff__mult,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( ( B2 != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B2)),C)
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1888_div__dvd__iff__mult,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( ( B2 != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),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),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1889_div__dvd__iff__mult,axiom,
    ! [B2: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1890_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B2,C))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1891_dvd__div__iff__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),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)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1892_dvd__div__iff__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1893_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
         => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),D)
           => ( ( divide6298287555418463151nteger(B2,A) = divide6298287555418463151nteger(D,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1894_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B2: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),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,B2),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1895_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B2: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1896_unit__div__eq__0__iff,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B2) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1897_unit__div__eq__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1898_unit__div__eq__0__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_1899_is__unit__power__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_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_1900_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_1901_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_1902_unit__imp__mod__eq__0,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_1903_unit__imp__mod__eq__0,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_1904_unit__imp__mod__eq__0,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_1905_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_1906_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1907_dvd__mult__cancel,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),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,Mb),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_1908_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_1909_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_1910_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_1911_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_1912_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_1913_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_1914_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_1915_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_1916_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_1917_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_1918_mod__greater__zero__iff__not__dvd,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1919_even__set__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se2793503036327961859nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1920_even__set__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1921_even__set__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        & ( Mb != zero_zero_nat ) ) ) ).

% even_set_bit_iff
tff(fact_1922_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se1345352211410354436nteger(Mb,A))
    <=> ~ ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1923_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se2159334234014336723it_int(Mb),A))
    <=> ~ ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1924_even__flip__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),A))
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        <=> ( Mb = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
tff(fact_1925_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se8260200283734997820nteger(Mb,A))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1926_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1927_even__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)
        | ( Mb = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
tff(fact_1928_dvd__triv__right,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A)) ).

% dvd_triv_right
tff(fact_1929_dvd__triv__right,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A)) ).

% dvd_triv_right
tff(fact_1930_dvd__triv__right,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)) ).

% dvd_triv_right
tff(fact_1931_dvd__triv__right,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)) ).

% dvd_triv_right
tff(fact_1932_dvd__triv__right,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)) ).

% dvd_triv_right
tff(fact_1933_dvd__mult__right,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ).

% dvd_mult_right
tff(fact_1934_dvd__mult__right,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),C) ) ).

% dvd_mult_right
tff(fact_1935_dvd__mult__right,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),C) ) ).

% dvd_mult_right
tff(fact_1936_dvd__mult__right,axiom,
    ! [A: nat,B2: 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),B2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ).

% dvd_mult_right
tff(fact_1937_dvd__mult__right,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ).

% dvd_mult_right
tff(fact_1938_mult__dvd__mono,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer,D: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( 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,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_1939_mult__dvd__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
     => ( 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,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_1940_mult__dvd__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,C),D)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_1941_mult__dvd__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( 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,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_1942_mult__dvd__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ).

% mult_dvd_mono
tff(fact_1943_dvd__triv__left,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) ).

% dvd_triv_left
tff(fact_1944_dvd__triv__left,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ).

% dvd_triv_left
tff(fact_1945_dvd__triv__left,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ).

% dvd_triv_left
tff(fact_1946_dvd__triv__left,axiom,
    ! [A: nat,B2: 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),B2)) ).

% dvd_triv_left
tff(fact_1947_dvd__triv__left,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ).

% dvd_triv_left
tff(fact_1948_dvd__mult__left,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)),C)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ).

% dvd_mult_left
tff(fact_1949_dvd__mult__left,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),C)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ).

% dvd_mult_left
tff(fact_1950_dvd__mult__left,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ).

% dvd_mult_left
tff(fact_1951_dvd__mult__left,axiom,
    ! [A: nat,B2: 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),B2)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_1952_dvd__mult__left,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_1953_dvd__mult2,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) ) ).

% dvd_mult2
tff(fact_1954_dvd__mult2,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),C)) ) ).

% dvd_mult2
tff(fact_1955_dvd__mult2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ).

% dvd_mult2
tff(fact_1956_dvd__mult2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) ) ).

% dvd_mult2
tff(fact_1957_dvd__mult2,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ).

% dvd_mult2
tff(fact_1958_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B2: 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,B2),C)) ) ).

% dvd_mult
tff(fact_1959_dvd__mult,axiom,
    ! [A: real,C: real,B2: 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,B2),C)) ) ).

% dvd_mult
tff(fact_1960_dvd__mult,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ).

% dvd_mult
tff(fact_1961_dvd__mult,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C)) ) ).

% dvd_mult
tff(fact_1962_dvd__mult,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ).

% dvd_mult
tff(fact_1963_dvd__def,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
    <=> ? [K3: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K3) ) ).

% dvd_def
tff(fact_1964_dvd__def,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
    <=> ? [K3: real] : A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K3) ) ).

% dvd_def
tff(fact_1965_dvd__def,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
    <=> ? [K3: rat] : A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),K3) ) ).

% dvd_def
tff(fact_1966_dvd__def,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
    <=> ? [K3: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K3) ) ).

% dvd_def
tff(fact_1967_dvd__def,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
    <=> ? [K3: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K3) ) ).

% dvd_def
tff(fact_1968_dvdI,axiom,
    ! [A: code_integer,B2: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A) ) ).

% dvdI
tff(fact_1969_dvdI,axiom,
    ! [A: real,B2: real,K: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A) ) ).

% dvdI
tff(fact_1970_dvdI,axiom,
    ! [A: rat,B2: rat,K: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A) ) ).

% dvdI
tff(fact_1971_dvdI,axiom,
    ! [A: nat,B2: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% dvdI
tff(fact_1972_dvdI,axiom,
    ! [A: int,B2: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B2),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% dvdI
tff(fact_1973_dvdE,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),K2) ) ).

% dvdE
tff(fact_1974_dvdE,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
     => ~ ! [K2: real] : A != aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),K2) ) ).

% dvdE
tff(fact_1975_dvdE,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
     => ~ ! [K2: rat] : A != aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),K2) ) ).

% dvdE
tff(fact_1976_dvdE,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),K2) ) ).

% dvdE
tff(fact_1977_dvdE,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B2),K2) ) ).

% dvdE
tff(fact_1978_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_1979_dvd__unit__imp__unit,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_1980_dvd__unit__imp__unit,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_1981_unit__imp__dvd,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A) ) ).

% unit_imp_dvd
tff(fact_1982_unit__imp__dvd,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ).

% unit_imp_dvd
tff(fact_1983_unit__imp__dvd,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ).

% unit_imp_dvd
tff(fact_1984_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,one_one_Code_integer),A) ).

% one_dvd
tff(fact_1985_one__dvd,axiom,
    ! [A: complex] : aa_complex_o(dvd_dvd_complex(one_one_complex),A) ).

% one_dvd
tff(fact_1986_one__dvd,axiom,
    ! [A: real] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,one_one_real),A) ).

% one_dvd
tff(fact_1987_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,one_one_rat),A) ).

% one_dvd
tff(fact_1988_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_1989_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_1990_dvd__add__right__iff,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),plus_p5714425477246183910nteger(B2,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1991_dvd__add__right__iff,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1992_dvd__add__right__iff,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1993_dvd__add__right__iff,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1994_dvd__add__right__iff,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_add_right_iff
tff(fact_1995_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: 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(B2,C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ) ).

% dvd_add_left_iff
tff(fact_1996_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B2: 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,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ) ).

% dvd_add_left_iff
tff(fact_1997_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2) ) ) ).

% dvd_add_left_iff
tff(fact_1998_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B2: 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,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ) ).

% dvd_add_left_iff
tff(fact_1999_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% dvd_add_left_iff
tff(fact_2000_dvd__add,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
     => ( 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(B2,C)) ) ) ).

% dvd_add
tff(fact_2001_dvd__add,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
     => ( 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,B2),C)) ) ) ).

% dvd_add
tff(fact_2002_dvd__add,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),C)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),C)) ) ) ).

% dvd_add
tff(fact_2003_dvd__add,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
     => ( 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,B2),C)) ) ) ).

% dvd_add
tff(fact_2004_dvd__add,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),C)) ) ) ).

% dvd_add
tff(fact_2005_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(C,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),minus_8373710615458151222nteger(B2,C)) ) ).

% dvd_diff_commute
tff(fact_2006_dvd__diff__commute,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B2))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),C)) ) ).

% dvd_diff_commute
tff(fact_2007_dvd__diff,axiom,
    ! [Xb: code_integer,Y: code_integer,Zb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Zb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),minus_8373710615458151222nteger(Y,Zb)) ) ) ).

% dvd_diff
tff(fact_2008_dvd__diff,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Zb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Zb)) ) ) ).

% dvd_diff
tff(fact_2009_dvd__diff,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Zb)
       => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(minus_minus_rat(Y),Zb)) ) ) ).

% dvd_diff
tff(fact_2010_dvd__diff,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Zb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Zb)) ) ) ).

% dvd_diff
tff(fact_2011_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B2: 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),B2)
       => ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B2,C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2012_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( aa_complex_o(dvd_dvd_complex(C),A)
     => ( aa_complex_o(dvd_dvd_complex(C),B2)
       => ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2013_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B2: 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),B2)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2014_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B2: 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),B2)
       => ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2015_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B2: 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),B2)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2016_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B2: 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),B2)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C) )
        <=> ( A = B2 ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2017_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( divide6298287555418463151nteger(A,C) = divide6298287555418463151nteger(B2,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),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2018_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),C) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
     => ( aa_complex_o(dvd_dvd_complex(C),A)
       => ( aa_complex_o(dvd_dvd_complex(C),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2019_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2020_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),C) = aa_rat_rat(divide_divide_rat(B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2021_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2022_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2023_div__div__div__same,axiom,
    ! [D: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
       => ( divide6298287555418463151nteger(divide6298287555418463151nteger(A,D),divide6298287555418463151nteger(B2,D)) = divide6298287555418463151nteger(A,B2) ) ) ) ).

% div_div_div_same
tff(fact_2024_div__div__div__same,axiom,
    ! [D: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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,B2),D)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ).

% div_div_div_same
tff(fact_2025_div__div__div__same,axiom,
    ! [D: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),D)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ) ).

% div_div_div_same
tff(fact_2026_dvd__power__same,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xb,Nb)),power_8256067586552552935nteger(Y,Nb)) ) ).

% dvd_power_same
tff(fact_2027_dvd__power__same,axiom,
    ! [Xb: nat,Y: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Nb)) ) ).

% dvd_power_same
tff(fact_2028_dvd__power__same,axiom,
    ! [Xb: real,Y: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb)) ) ).

% dvd_power_same
tff(fact_2029_dvd__power__same,axiom,
    ! [Xb: int,Y: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Nb)) ) ).

% dvd_power_same
tff(fact_2030_dvd__power__same,axiom,
    ! [Xb: complex,Y: complex,Nb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Y)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Nb)) ) ).

% dvd_power_same
tff(fact_2031_dvd__mod__iff,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2032_dvd__mod__iff,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2033_dvd__mod__iff,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_iff
tff(fact_2034_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2035_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2036_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),modulo364778990260209775nteger(A,B2))
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2037_dvd__mod,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_2038_dvd__mod,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Mb)
     => ( 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(aa_int_fun_int_int(modulo_modulo_int,Mb),Nb)) ) ) ).

% dvd_mod
tff(fact_2039_dvd__mod,axiom,
    ! [K: code_integer,Mb: code_integer,Nb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,K),Mb)
     => ( 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(Mb,Nb)) ) ) ).

% dvd_mod
tff(fact_2040_mod__mod__cancel,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) ) ) ).

% mod_mod_cancel
tff(fact_2041_mod__mod__cancel,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) ) ) ).

% mod_mod_cancel
tff(fact_2042_mod__mod__cancel,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( modulo364778990260209775nteger(modulo364778990260209775nteger(A,B2),C) = modulo364778990260209775nteger(A,C) ) ) ).

% mod_mod_cancel
tff(fact_2043_dvd__diff__nat,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
     => ( 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,Mb),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_2044_zero__le,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb) ).

% zero_le
tff(fact_2045_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_2046_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_2047_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_2048_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_2049_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_2050_gr__implies__not__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_2051_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_2052_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_2053_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_2054_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_2055_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_2056_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_2057_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D22)
       => ? [E2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D1)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2058_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D1)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),D22)
       => ? [E2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D1)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,E2),D22) ) ) ) ).

% field_lbound_gt_zero
tff(fact_2059_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != numera6690914467698888265omplex(Nb) ).

% zero_neq_numeral
tff(fact_2060_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_real != aa_num_real(numeral_numeral_real,Nb) ).

% zero_neq_numeral
tff(fact_2061_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_rat != aa_num_rat(numeral_numeral_rat,Nb) ).

% zero_neq_numeral
tff(fact_2062_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_nat != aa_num_nat(numeral_numeral_nat,Nb) ).

% zero_neq_numeral
tff(fact_2063_zero__neq__numeral,axiom,
    ! [Nb: num] : zero_zero_int != aa_num_int(numeral_numeral_int,Nb) ).

% zero_neq_numeral
tff(fact_2064_mult__right__cancel,axiom,
    ! [C: real,A: real,B2: 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,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2065_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2066_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2067_mult__right__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),C) )
      <=> ( A = B2 ) ) ) ).

% mult_right_cancel
tff(fact_2068_mult__left__cancel,axiom,
    ! [C: real,A: real,B2: 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),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2069_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2070_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B2: 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),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2071_mult__left__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) )
      <=> ( A = B2 ) ) ) ).

% mult_left_cancel
tff(fact_2072_no__zero__divisors,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) != zero_zero_real ) ) ) ).

% no_zero_divisors
tff(fact_2073_no__zero__divisors,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_2074_no__zero__divisors,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B2 != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_2075_no__zero__divisors,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( ( B2 != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_2076_divisors__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B2 = zero_zero_real ) ) ) ).

% divisors_zero
tff(fact_2077_divisors__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B2 = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_2078_divisors__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B2 = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_2079_divisors__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B2 = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_2080_mult__not__zero,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B2 != zero_zero_real ) ) ) ).

% mult_not_zero
tff(fact_2081_mult__not__zero,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B2 != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_2082_mult__not__zero,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B2 != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_2083_mult__not__zero,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B2 != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_2084_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
tff(fact_2085_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
tff(fact_2086_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_2087_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_2088_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_2089_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_2090_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2091_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_2092_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_2093_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2094_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_2095_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_2096_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_2097_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2098_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_2099_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2100_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_2101_verit__sum__simplify,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% verit_sum_simplify
tff(fact_2102_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_2103_verit__sum__simplify,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% verit_sum_simplify
tff(fact_2104_eq__iff__diff__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = zero_zero_real ) ) ).

% eq_iff_diff_eq_0
tff(fact_2105_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_rat(minus_minus_rat(A),B2) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_2106_eq__iff__diff__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_2107_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_2108_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_2109_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_2110_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_2111_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_2112_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero_nat )
     => ? [M3: nat] : Nb = aa_nat_nat(suc,M3) ) ).

% not0_implies_Suc
tff(fact_2113_Zero__not__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_not_Suc
tff(fact_2114_Zero__neq__Suc,axiom,
    ! [Mb: nat] : zero_zero_nat != aa_nat_nat(suc,Mb) ).

% Zero_neq_Suc
tff(fact_2115_Suc__neq__Zero,axiom,
    ! [Mb: nat] : aa_nat_nat(suc,Mb) != zero_zero_nat ).

% Suc_neq_Zero
tff(fact_2116_zero__induct,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( aa_nat_o(P,K)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,aa_nat_nat(suc,N2))
           => aa_nat_o(P,N2) )
       => aa_nat_o(P,zero_zero_nat) ) ) ).

% zero_induct
tff(fact_2117_diff__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: 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,Mb),Nb) ) ) ) ).

% diff_induct
tff(fact_2118_nat__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => aa_nat_o(P,aa_nat_nat(suc,N2)) )
       => aa_nat_o(P,Nb) ) ) ).

% nat_induct
tff(fact_2119_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] : Y != aa_nat_nat(suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_2120_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat = aa_nat_nat(suc,X2) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
tff(fact_2121_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero_nat != aa_nat_nat(suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_2122_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa_nat_nat(suc,Nat2) != zero_zero_nat ).

% old.nat.distinct(2)
tff(fact_2123_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] : zero_zero_nat != aa_nat_nat(suc,X2) ).

% nat.distinct(1)
tff(fact_2124_infinite__descent0,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( ~ aa_nat_o(P,N2)
             => ? [M4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N2)
                  & ~ aa_nat_o(P,M4) ) ) )
       => aa_nat_o(P,Nb) ) ) ).

% infinite_descent0
tff(fact_2125_gr__implies__not0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => ( Nb != zero_zero_nat ) ) ).

% gr_implies_not0
tff(fact_2126_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% less_zeroE
tff(fact_2127_not__less0,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),zero_zero_nat) ).

% not_less0
tff(fact_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_add__eq__self__zero,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = Mb )
     => ( Nb = zero_zero_nat ) ) ).

% add_eq_self_zero
tff(fact_2136_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_2137_diffs0__imp__equal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb) = zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb) = zero_zero_nat )
       => ( Mb = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_2138_minus__nat_Odiff__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),zero_zero_nat) = Mb ).

% minus_nat.diff_0
tff(fact_2139_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
    <=> ( ( K = zero_zero_nat )
        | ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_2140_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_2141_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_2142_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_2143_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_2144_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_2145_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_2146_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(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B4 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B4) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B4) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B4) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2147_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) = divide6298287555418463151nteger(one_one_Code_integer,B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2148_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2149_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2150_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A)) = divide6298287555418463151nteger(one_one_Code_integer,B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2151_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2152_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B2: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B2) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2153_dvd__power__iff,axiom,
    ! [Xb: code_integer,Mb: nat,Nb: nat] :
      ( ( Xb != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xb,Mb)),power_8256067586552552935nteger(Xb,Nb))
      <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2154_dvd__power__iff,axiom,
    ! [Xb: nat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2155_dvd__power__iff,axiom,
    ! [Xb: int,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Mb)),aa_nat_int(power_power_int(Xb),Nb))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ) ).

% dvd_power_iff
tff(fact_2156_dvd__power,axiom,
    ! [Nb: nat,Xb: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_Code_integer ) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),power_8256067586552552935nteger(Xb,Nb)) ) ).

% dvd_power
tff(fact_2157_dvd__power,axiom,
    ! [Nb: nat,Xb: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_rat ) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_nat_rat(power_power_rat(Xb),Nb)) ) ).

% dvd_power
tff(fact_2158_dvd__power,axiom,
    ! [Nb: nat,Xb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)) ) ).

% dvd_power
tff(fact_2159_dvd__power,axiom,
    ! [Nb: nat,Xb: real] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_real ) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) ) ).

% dvd_power
tff(fact_2160_dvd__power,axiom,
    ! [Nb: nat,Xb: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_nat_int(power_power_int(Xb),Nb)) ) ).

% dvd_power
tff(fact_2161_dvd__power,axiom,
    ! [Nb: nat,Xb: complex] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
        | ( Xb = one_one_complex ) )
     => aa_complex_o(dvd_dvd_complex(Xb),aa_nat_complex(power_power_complex(Xb),Nb)) ) ).

% dvd_power
tff(fact_2162_enat__0__less__mult__iff,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),times_7803423173614009249d_enat(Mb,Nb))
    <=> ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Mb)
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_2163_not__iless0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),zero_z5237406670263579293d_enat) ).

% not_iless0
tff(fact_2164_iadd__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        & ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
tff(fact_2165_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),zero_z5237406670263579293d_enat)
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
tff(fact_2166_i0__lb,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(zero_z5237406670263579293d_enat),Nb) ).

% i0_lb
tff(fact_2167_dvd__mult__cancel1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_2168_dvd__mult__cancel2,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)),Mb)
      <=> ( Nb = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_2169_power__eq__imp__eq__base,axiom,
    ! [A: real,Nb: nat,B2: 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,B2),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),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2170_power__eq__imp__eq__base,axiom,
    ! [A: rat,Nb: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B2),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),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2171_power__eq__imp__eq__base,axiom,
    ! [A: nat,Nb: nat,B2: 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,B2),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),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2172_power__eq__imp__eq__base,axiom,
    ! [A: int,Nb: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B2),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),B2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
           => ( A = B2 ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_2173_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: real,B2: 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),B2)
         => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2174_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: rat,B2: 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),B2)
         => ( ( aa_nat_rat(power_power_rat(A),Nb) = aa_nat_rat(power_power_rat(B2),Nb) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2175_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: nat,B2: 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),B2)
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),Nb) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2176_power__eq__iff__eq__base,axiom,
    ! [Nb: nat,A: int,B2: 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),B2)
         => ( ( aa_nat_int(power_power_int(A),Nb) = aa_nat_int(power_power_int(B2),Nb) )
          <=> ( A = B2 ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_2177_int__power__div__base,axiom,
    ! [Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(K),Mb)),K) = aa_nat_int(power_power_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,zero_zero_nat))) ) ) ) ).

% int_power_div_base
tff(fact_2178_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_2179_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_2180_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_2181_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri6519982836138164636nteger(Mb,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_2182_even__signed__take__bit__iff,axiom,
    ! [Mb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(Mb),A))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) ) ).

% even_signed_take_bit_iff
tff(fact_2183_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)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),Nb) = A ) ) ) ) ).

% realpow_pos_nth
tff(fact_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2190_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_2191_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_2192_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_2193_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_2194_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_2195_is__unit__mult__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)),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,B2),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_2196_is__unit__mult__iff,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),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,B2),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_2197_is__unit__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),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,B2),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_2198_dvd__mult__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2199_dvd__mult__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2200_dvd__mult__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_2201_mult__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2202_mult__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2203_mult__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_2204_dvd__mult__unit__iff_H,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2205_dvd__mult__unit__iff_H,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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,B2),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2206_dvd__mult__unit__iff_H,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2207_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B2: 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),B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2208_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B2: 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),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2209_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2210_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B2: 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),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2211_unit__mult__left__cancel,axiom,
    ! [A: nat,B2: 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),B2) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2212_unit__mult__left__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_2213_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B2: 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,B2),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2214_unit__mult__right__cancel,axiom,
    ! [A: nat,B2: 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,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2215_unit__mult__right__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B2),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_2216_dvd__div__mult,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B2,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),A),C) ) ) ).

% dvd_div_mult
tff(fact_2217_dvd__div__mult,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),A)),C) ) ) ).

% dvd_div_mult
tff(fact_2218_dvd__div__mult,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),C) ) ) ).

% dvd_div_mult
tff(fact_2219_div__mult__swap,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B2,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) ) ) ).

% div_mult_swap
tff(fact_2220_div__mult__swap,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) ) ) ).

% div_mult_swap
tff(fact_2221_div__mult__swap,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ) ) ).

% div_mult_swap
tff(fact_2222_div__div__eq__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B2,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2223_div__div__eq__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2224_div__div__eq__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ) ).

% div_div_eq_right
tff(fact_2225_dvd__div__mult2__eq,axiom,
    ! [B2: 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,B2),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B2),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2226_dvd__div__mult2__eq,axiom,
    ! [B2: 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,B2),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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2227_dvd__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_2228_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(B2,C)) ) ).

% dvd_mult_imp_div
tff(fact_2229_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B2: 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)),B2)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),C)) ) ).

% dvd_mult_imp_div
tff(fact_2230_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ).

% dvd_mult_imp_div
tff(fact_2231_div__mult__div__if__dvd,axiom,
    ! [B2: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),divide6298287555418463151nteger(C,D)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2232_div__mult__div__if__dvd,axiom,
    ! [B2: nat,A: nat,D: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2)),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,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2233_div__mult__div__if__dvd,axiom,
    ! [B2: int,A: int,D: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2234_unit__div__cancel,axiom,
    ! [A: code_integer,B2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B2,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_2235_unit__div__cancel,axiom,
    ! [A: nat,B2: 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,B2),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_2236_unit__div__cancel,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B2 = C ) ) ) ).

% unit_div_cancel
tff(fact_2237_div__unit__dvd__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,divide6298287555418463151nteger(A,B2)),C)
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2238_div__unit__dvd__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2239_div__unit__dvd__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_2240_dvd__div__unit__iff,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),divide6298287555418463151nteger(C,B2))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2241_dvd__div__unit__iff,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2242_dvd__div__unit__iff,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_2243_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),B2)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B2),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B2,C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2244_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),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,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2245_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2246_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),A)
     => ( divide6298287555418463151nteger(plus_p5714425477246183910nteger(A,B2),C) = plus_p5714425477246183910nteger(divide6298287555418463151nteger(A,C),divide6298287555418463151nteger(B2,C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2247_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B2: 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),B2)),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,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2248_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),C)),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2249_div__power,axiom,
    ! [B2: code_integer,A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ( power_8256067586552552935nteger(divide6298287555418463151nteger(A,B2),Nb) = divide6298287555418463151nteger(power_8256067586552552935nteger(A,Nb),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% div_power
tff(fact_2250_div__power,axiom,
    ! [B2: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),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,B2),Nb)) ) ) ).

% div_power
tff(fact_2251_div__power,axiom,
    ! [B2: int,A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),Nb) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Nb)),aa_nat_int(power_power_int(B2),Nb)) ) ) ).

% div_power
tff(fact_2252_power__strict__mono,axiom,
    ! [A: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2253_power__strict__mono,axiom,
    ! [A: rat,B2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( 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(B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2254_power__strict__mono,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2255_power__strict__mono,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( 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(B2),Nb)) ) ) ) ).

% power_strict_mono
tff(fact_2256_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A,Mb)),power_8256067586552552935nteger(A,Nb)) ) ).

% le_imp_power_dvd
tff(fact_2257_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2258_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2259_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2260_le__imp__power__dvd,axiom,
    ! [Mb: nat,Nb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ).

% le_imp_power_dvd
tff(fact_2261_power__le__dvd,axiom,
    ! [A: code_integer,Nb: nat,B2: code_integer,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A,Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(A,Mb)),B2) ) ) ).

% power_le_dvd
tff(fact_2262_power__le__dvd,axiom,
    ! [A: nat,Nb: nat,B2: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),B2) ) ) ).

% power_le_dvd
tff(fact_2263_power__le__dvd,axiom,
    ! [A: real,Nb: nat,B2: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),B2) ) ) ).

% power_le_dvd
tff(fact_2264_power__le__dvd,axiom,
    ! [A: int,Nb: nat,B2: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),Mb)),B2) ) ) ).

% power_le_dvd
tff(fact_2265_power__le__dvd,axiom,
    ! [A: complex,Nb: nat,B2: complex,Mb: nat] :
      ( aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Nb)),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(A),Mb)),B2) ) ) ).

% power_le_dvd
tff(fact_2266_dvd__power__le,axiom,
    ! [Xb: code_integer,Y: code_integer,Nb: nat,Mb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(Xb,Nb)),power_8256067586552552935nteger(Y,Mb)) ) ) ).

% dvd_power_le
tff(fact_2267_dvd__power__le,axiom,
    ! [Xb: nat,Y: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_2268_dvd__power__le,axiom,
    ! [Xb: real,Y: real,Nb: nat,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_2269_dvd__power__le,axiom,
    ! [Xb: int,Y: int,Nb: nat,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(Xb),Nb)),aa_nat_int(power_power_int(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_2270_dvd__power__le,axiom,
    ! [Xb: complex,Y: complex,Nb: nat,Mb: nat] :
      ( aa_complex_o(dvd_dvd_complex(Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => aa_complex_o(dvd_dvd_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(Y),Mb)) ) ) ).

% dvd_power_le
tff(fact_2271_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),C) )
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ).

% mod_eq_dvd_iff
tff(fact_2272_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B2,C) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,C),minus_8373710615458151222nteger(A,B2)) ) ).

% mod_eq_dvd_iff
tff(fact_2273_dvd__minus__mod,axiom,
    ! [B2: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2))) ).

% dvd_minus_mod
tff(fact_2274_dvd__minus__mod,axiom,
    ! [B2: int,A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))) ).

% dvd_minus_mod
tff(fact_2275_dvd__minus__mod,axiom,
    ! [B2: code_integer,A: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B2))) ).

% dvd_minus_mod
tff(fact_2276_dvd__minus__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ) ).

% dvd_minus_self
tff(fact_2277_dvd__diffD,axiom,
    ! [K: nat,Mb: 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,Mb),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),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb) ) ) ) ).

% dvd_diffD
tff(fact_2278_dvd__diffD1,axiom,
    ! [K: nat,Mb: 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,Mb),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_2279_less__eq__dvd__minus,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% less_eq_dvd_minus
tff(fact_2280_dbl__def,axiom,
    ! [Xb: real] : aa_real_real(neg_numeral_dbl_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb) ).

% dbl_def
tff(fact_2281_dbl__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_numeral_dbl_rat,Xb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb) ).

% dbl_def
tff(fact_2282_dbl__def,axiom,
    ! [Xb: int] : aa_int_int(neg_numeral_dbl_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb) ).

% dbl_def
tff(fact_2283_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_2284_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_2285_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_2286_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_2287_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_2288_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_2289_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_2290_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_2291_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2292_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_2295_zero__le__mult__iff,axiom,
    ! [A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_real) ) ) ) ).

% zero_le_mult_iff
tff(fact_2296_zero__le__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_2297_zero__le__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_2298_mult__nonneg__nonpos2,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2299_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2300_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B2: 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,B2),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,B2),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2301_mult__nonneg__nonpos2,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_2302_mult__nonpos__nonneg,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_nonpos_nonneg
tff(fact_2303_mult__nonpos__nonneg,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2304_mult__nonpos__nonneg,axiom,
    ! [A: nat,B2: 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),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_2305_mult__nonpos__nonneg,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_2306_mult__nonneg__nonpos,axiom,
    ! [A: real,B2: 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,B2),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),B2)),zero_zero_real) ) ) ).

% mult_nonneg_nonpos
tff(fact_2307_mult__nonneg__nonpos,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2308_mult__nonneg__nonpos,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_2309_mult__nonneg__nonpos,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_2310_mult__nonneg__nonneg,axiom,
    ! [A: real,B2: 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),B2)
       => 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),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2311_mult__nonneg__nonneg,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2312_mult__nonneg__nonneg,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2313_mult__nonneg__nonneg,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_nonneg_nonneg
tff(fact_2314_split__mult__neg__le,axiom,
    ! [A: real,B2: 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,B2),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),B2) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ).

% split_mult_neg_le
tff(fact_2315_split__mult__neg__le,axiom,
    ! [A: rat,B2: 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,B2),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),B2) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_2316_split__mult__neg__le,axiom,
    ! [A: nat,B2: 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,B2),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),B2) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_2317_split__mult__neg__le,axiom,
    ! [A: int,B2: 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,B2),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),B2) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_2318_mult__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2319_mult__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2320_mult__le__0__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_le_0_iff
tff(fact_2321_mult__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2322_mult__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2323_mult__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2324_mult__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_right_mono
tff(fact_2325_mult__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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,B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2326_mult__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2327_mult__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_right_mono_neg
tff(fact_2328_mult__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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),B2)) ) ) ).

% mult_left_mono
tff(fact_2329_mult__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% mult_left_mono
tff(fact_2330_mult__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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),B2)) ) ) ).

% mult_left_mono
tff(fact_2331_mult__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_left_mono
tff(fact_2332_mult__nonpos__nonpos,axiom,
    ! [A: real,B2: 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,B2),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),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2333_mult__nonpos__nonpos,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2334_mult__nonpos__nonpos,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_nonpos_nonpos
tff(fact_2335_mult__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2336_mult__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2337_mult__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_left_mono_neg
tff(fact_2338_split__mult__pos__le,axiom,
    ! [A: real,B2: 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),B2) )
        | ( 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,B2),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),B2)) ) ).

% split_mult_pos_le
tff(fact_2339_split__mult__pos__le,axiom,
    ! [A: rat,B2: 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),B2) )
        | ( 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,B2),zero_zero_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ).

% split_mult_pos_le
tff(fact_2340_split__mult__pos__le,axiom,
    ! [A: int,B2: 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),B2) )
        | ( 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,B2),zero_zero_int) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ).

% split_mult_pos_le
tff(fact_2341_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_2342_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_2343_zero__le__square,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),A)) ).

% zero_le_square
tff(fact_2344_mult__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2345_mult__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2346_mult__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2347_mult__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono'
tff(fact_2348_mult__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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),B2)
         => ( 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,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2349_mult__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2350_mult__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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),B2)
         => ( 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,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2351_mult__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( 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),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_mono
tff(fact_2352_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_2353_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_2354_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_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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_2370_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_2371_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_2372_add__decreasing,axiom,
    ! [A: real,C: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B2) ) ) ).

% add_decreasing
tff(fact_2373_add__decreasing,axiom,
    ! [A: rat,C: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B2) ) ) ).

% add_decreasing
tff(fact_2374_add__decreasing,axiom,
    ! [A: nat,C: nat,B2: 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),B2)
       => 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)),B2) ) ) ).

% add_decreasing
tff(fact_2375_add__decreasing,axiom,
    ! [A: int,C: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2) ) ) ).

% add_decreasing
tff(fact_2376_add__increasing,axiom,
    ! [A: real,B2: 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,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing
tff(fact_2377_add__increasing,axiom,
    ! [A: rat,B2: 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,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_2378_add__increasing,axiom,
    ! [A: nat,B2: 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,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_2379_add__increasing,axiom,
    ! [A: int,B2: 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,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_2380_add__decreasing2,axiom,
    ! [C: real,A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),B2) ) ) ).

% add_decreasing2
tff(fact_2381_add__decreasing2,axiom,
    ! [C: rat,A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B2) ) ) ).

% add_decreasing2
tff(fact_2382_add__decreasing2,axiom,
    ! [C: nat,A: nat,B2: 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),B2)
       => 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)),B2) ) ) ).

% add_decreasing2
tff(fact_2383_add__decreasing2,axiom,
    ! [C: int,A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B2) ) ) ).

% add_decreasing2
tff(fact_2384_add__increasing2,axiom,
    ! [C: real,B2: 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,B2),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_increasing2
tff(fact_2385_add__increasing2,axiom,
    ! [C: rat,B2: 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,B2),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_2386_add__increasing2,axiom,
    ! [C: nat,B2: 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,B2),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_2387_add__increasing2,axiom,
    ! [C: int,B2: 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,B2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_2388_add__nonneg__nonneg,axiom,
    ! [A: real,B2: 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),B2)
       => 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),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2389_add__nonneg__nonneg,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2390_add__nonneg__nonneg,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2391_add__nonneg__nonneg,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_nonneg_nonneg
tff(fact_2392_add__nonpos__nonpos,axiom,
    ! [A: real,B2: 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,B2),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),B2)),zero_zero_real) ) ) ).

% add_nonpos_nonpos
tff(fact_2393_add__nonpos__nonpos,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2394_add__nonpos__nonpos,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2395_add__nonpos__nonpos,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2396_add__nonneg__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2397_add__nonneg__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2398_add__nonneg__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2399_add__nonneg__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2400_add__nonpos__eq__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y) = zero_zero_real )
        <=> ( ( Xb = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2401_add__nonpos__eq__0__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y) = zero_zero_rat )
        <=> ( ( Xb = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2402_add__nonpos__eq__0__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),zero_zero_nat)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = zero_zero_nat )
        <=> ( ( Xb = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2403_add__nonpos__eq__0__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),zero_zero_int)
       => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) = zero_zero_int )
        <=> ( ( Xb = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2404_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2405_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2406_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2407_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_2408_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2409_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2410_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_2411_mult__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2412_mult__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2413_mult__strict__right__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2414_mult__strict__right__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_strict_right_mono
tff(fact_2415_mult__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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,B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2416_mult__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2417_mult__strict__right__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_2418_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2419_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2420_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_2421_mult__strict__left__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2422_mult__strict__left__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2423_mult__strict__left__mono,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2424_mult__strict__left__mono,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_strict_left_mono
tff(fact_2425_mult__strict__left__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2426_mult__strict__left__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2427_mult__strict__left__mono__neg,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_2428_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: 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),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2429_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2430_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% mult_less_cancel_left_pos
tff(fact_2431_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: 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),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2432_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2433_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_2434_zero__less__mult__pos2,axiom,
    ! [B2: 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,B2),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),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2435_zero__less__mult__pos2,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2436_zero__less__mult__pos2,axiom,
    ! [B2: 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,B2),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),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2437_zero__less__mult__pos2,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) ) ) ).

% zero_less_mult_pos2
tff(fact_2438_zero__less__mult__pos,axiom,
    ! [A: real,B2: 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),B2))
     => ( 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),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2439_zero__less__mult__pos,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
     => ( 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),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2440_zero__less__mult__pos,axiom,
    ! [A: nat,B2: 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),B2))
     => ( 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),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2441_zero__less__mult__pos,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
     => ( 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),B2) ) ) ).

% zero_less_mult_pos
tff(fact_2442_zero__less__mult__iff,axiom,
    ! [A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_real) ) ) ) ).

% zero_less_mult_iff
tff(fact_2443_zero__less__mult__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_2444_zero__less__mult__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_2445_mult__pos__neg2,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),A)),zero_zero_real) ) ) ).

% mult_pos_neg2
tff(fact_2446_mult__pos__neg2,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_2447_mult__pos__neg2,axiom,
    ! [A: nat,B2: 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,B2),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,B2),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_2448_mult__pos__neg2,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_2449_mult__pos__pos,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_2450_mult__pos__pos,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_2451_mult__pos__pos,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% mult_pos_pos
tff(fact_2452_mult__pos__pos,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_pos_pos
tff(fact_2453_mult__pos__neg,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_pos_neg
tff(fact_2454_mult__pos__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_2455_mult__pos__neg,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_2456_mult__pos__neg,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_2457_mult__neg__pos,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real) ) ) ).

% mult_neg_pos
tff(fact_2458_mult__neg__pos,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_2459_mult__neg__pos,axiom,
    ! [A: nat,B2: 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),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_2460_mult__neg__pos,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_2461_mult__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2462_mult__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2463_mult__less__0__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),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,B2),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),B2) ) ) ) ).

% mult_less_0_iff
tff(fact_2464_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_2465_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_2466_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_2467_mult__neg__neg,axiom,
    ! [A: real,B2: 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,B2),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),B2)) ) ) ).

% mult_neg_neg
tff(fact_2468_mult__neg__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ) ) ).

% mult_neg_neg
tff(fact_2469_mult__neg__neg,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ) ).

% mult_neg_neg
tff(fact_2470_le__iff__diff__le__0,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),zero_zero_real) ) ).

% le_iff_diff_le_0
tff(fact_2471_le__iff__diff__le__0,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),B2)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_2472_le__iff__diff__le__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_add__neg__neg,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_neg_neg
tff(fact_2486_add__neg__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2487_add__neg__neg,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2488_add__neg__neg,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2489_add__pos__pos,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_pos_pos
tff(fact_2490_add__pos__pos,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_pos_pos
tff(fact_2491_add__pos__pos,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% add_pos_pos
tff(fact_2492_add__pos__pos,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_pos_pos
tff(fact_2493_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ ! [C3: nat] :
            ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2494_pos__add__strict,axiom,
    ! [A: real,B2: 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,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% pos_add_strict
tff(fact_2495_pos__add__strict,axiom,
    ! [A: rat,B2: 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,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2496_pos__add__strict,axiom,
    ! [A: nat,B2: 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,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2497_pos__add__strict,axiom,
    ! [A: int,B2: 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,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_2498_add__less__zeroD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real) ) ) ).

% add_less_zeroD
tff(fact_2499_add__less__zeroD,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat) ) ) ).

% add_less_zeroD
tff(fact_2500_add__less__zeroD,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y)),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),zero_zero_int) ) ) ).

% add_less_zeroD
tff(fact_2501_divide__le__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),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,B2),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),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_2502_divide__le__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(A),B2)),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,B2),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),B2) ) ) ) ).

% divide_le_0_iff
tff(fact_2503_divide__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),C)) ) ) ).

% divide_right_mono
tff(fact_2504_divide__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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(B2),C)) ) ) ).

% divide_right_mono
tff(fact_2505_zero__le__divide__iff,axiom,
    ! [A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_real) ) ) ) ).

% zero_le_divide_iff
tff(fact_2506_zero__le__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_rat) ) ) ) ).

% zero_le_divide_iff
tff(fact_2507_divide__nonneg__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2508_divide__nonneg__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_nonneg
tff(fact_2509_divide__nonneg__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_nonpos
tff(fact_2510_divide__nonneg__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_nonpos
tff(fact_2511_divide__nonpos__nonneg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_nonneg
tff(fact_2512_divide__nonpos__nonneg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_nonneg
tff(fact_2513_divide__nonpos__nonpos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2514_divide__nonpos__nonpos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonpos_nonpos
tff(fact_2515_divide__right__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),C)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2516_divide__right__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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(B2),C)),aa_rat_rat(divide_divide_rat(A),C)) ) ) ).

% divide_right_mono_neg
tff(fact_2517_less__iff__diff__less__0,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)),zero_zero_real) ) ).

% less_iff_diff_less_0
tff(fact_2518_less__iff__diff__less__0,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),B2)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_2519_less__iff__diff__less__0,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_2520_divide__strict__right__mono__neg,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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,B2),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2521_divide__strict__right__mono__neg,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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(B2),C)) ) ) ).

% divide_strict_right_mono_neg
tff(fact_2522_divide__strict__right__mono,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2523_divide__strict__right__mono,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( 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(B2),C)) ) ) ).

% divide_strict_right_mono
tff(fact_2524_zero__less__divide__iff,axiom,
    ! [A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_real) ) ) ) ).

% zero_less_divide_iff
tff(fact_2525_zero__less__divide__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(A),B2))
    <=> ( ( 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),B2) )
        | ( 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,B2),zero_zero_rat) ) ) ) ).

% zero_less_divide_iff
tff(fact_2526_divide__less__cancel,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) )
        & ( 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,B2),A) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
tff(fact_2527_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B2: 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(B2),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),B2) )
        & ( 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,B2),A) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
tff(fact_2528_divide__less__0__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)),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,B2),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),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_2529_divide__less__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(A),B2)),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,B2),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),B2) ) ) ) ).

% divide_less_0_iff
tff(fact_2530_divide__pos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_2531_divide__pos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_pos_pos
tff(fact_2532_divide__pos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_pos_neg
tff(fact_2533_divide__pos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_pos_neg
tff(fact_2534_divide__neg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_neg_pos
tff(fact_2535_divide__neg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_neg_pos
tff(fact_2536_divide__neg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_2537_divide__neg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_neg_neg
tff(fact_2538_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_2539_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_2540_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_2541_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_2542_power__mono,axiom,
    ! [A: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),Nb)) ) ) ).

% power_mono
tff(fact_2543_power__mono,axiom,
    ! [A: rat,B2: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( 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(B2),Nb)) ) ) ).

% power_mono
tff(fact_2544_power__mono,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),Nb)) ) ) ).

% power_mono
tff(fact_2545_power__mono,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( 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(B2),Nb)) ) ) ).

% power_mono
tff(fact_2546_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_2547_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_2548_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_2549_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_2550_frac__eq__eq,axiom,
    ! [Y: complex,Zb: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Zb != zero_zero_complex )
       => ( ( aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(divide1717551699836669952omplex(W),Zb) )
        <=> ( aa_complex_complex(times_times_complex(Xb),Zb) = aa_complex_complex(times_times_complex(W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2551_frac__eq__eq,axiom,
    ! [Y: real,Zb: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Zb != zero_zero_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Zb) )
        <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2552_frac__eq__eq,axiom,
    ! [Y: rat,Zb: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Zb != zero_zero_rat )
       => ( ( aa_rat_rat(divide_divide_rat(Xb),Y) = aa_rat_rat(divide_divide_rat(W),Zb) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2553_divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A )
    <=> $ite(C != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% divide_eq_eq
tff(fact_2554_divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) = A )
    <=> $ite(C != zero_zero_real,B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% divide_eq_eq
tff(fact_2555_divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C) = A )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2556_eq__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq
tff(fact_2557_eq__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B2,A = zero_zero_real) ) ).

% eq_divide_eq
tff(fact_2558_eq__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = aa_rat_rat(divide_divide_rat(B2),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2559_divide__eq__imp,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B2 = aa_complex_complex(times_times_complex(A),C) )
       => ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2560_divide__eq__imp,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2561_divide__eq__imp,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( aa_rat_rat(divide_divide_rat(B2),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2562_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(times_times_complex(A),C) = B2 )
       => ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2563_eq__divide__imp,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B2 )
       => ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2564_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B2 )
       => ( A = aa_rat_rat(divide_divide_rat(B2),C) ) ) ) ).

% eq_divide_imp
tff(fact_2565_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B2: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = A )
      <=> ( B2 = aa_complex_complex(times_times_complex(A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2566_nonzero__divide__eq__eq,axiom,
    ! [C: real,B2: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) = A )
      <=> ( B2 = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2567_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B2: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(B2),C) = A )
      <=> ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2568_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B2: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
      <=> ( aa_complex_complex(times_times_complex(A),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2569_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B2: real] :
      ( ( C != zero_zero_real )
     => ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2570_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(divide_divide_rat(B2),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B2 ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2571_right__inverse__eq,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = one_one_complex )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2572_right__inverse__eq,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = one_one_real )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2573_right__inverse__eq,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(divide_divide_rat(A),B2) = one_one_rat )
      <=> ( A = B2 ) ) ) ).

% right_inverse_eq
tff(fact_2574_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_2575_pos__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_2576_neg__zdiv__mult__2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_2577_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B2: 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,B2)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2578_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B2: 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),B2)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2579_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),A) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2580_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2581_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2582_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,B2)),B2) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2583_power__0,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2584_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_2585_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_2586_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2587_power__0,axiom,
    ! [A: complex] : aa_nat_complex(power_power_complex(A),zero_zero_nat) = one_one_complex ).

% power_0
tff(fact_2588_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2589_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = A )
    <=> ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2590_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B2) = A )
    <=> ( divide6298287555418463151nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2591_less__Suc__eq__0__disj,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(suc,Nb))
    <=> ( ( Mb = zero_zero_nat )
        | ? [J3: nat] :
            ( ( Mb = aa_nat_nat(suc,J3) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_2592_gr0__implies__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_implies_Suc
tff(fact_2593_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
         => aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
           => aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_2594_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
    <=> ? [M6: nat] : Nb = aa_nat_nat(suc,M6) ) ).

% gr0_conv_Suc
tff(fact_2595_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),aa_nat_nat(suc,Nb))
          & aa_nat_o(P,I2) )
    <=> ( aa_nat_o(P,zero_zero_nat)
        | ? [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),Nb)
            & aa_nat_o(P,aa_nat_nat(suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_2596_add__is__1,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% add_is_1
tff(fact_2597_one__is__add,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(suc,zero_zero_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) )
    <=> ( ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
          & ( Nb = zero_zero_nat ) )
        | ( ( Mb = zero_zero_nat )
          & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ) ).

% one_is_add
tff(fact_2598_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)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_2599_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ? [K2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K2)
          & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_2600_diff__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Mb) ) ) ).

% diff_less
tff(fact_2601_nat__mult__less__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_2602_nat__mult__eq__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb) )
      <=> ( Mb = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_2603_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),I)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),J)) ) ) ).

% mult_less_mono2
tff(fact_2604_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,I),K)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,J),K)) ) ) ).

% mult_less_mono1
tff(fact_2605_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_2606_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = zero_zero_nat )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_2607_diff__add__0,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = zero_zero_nat ).

% diff_add_0
tff(fact_2608_nat__power__less__imp__less,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),I)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_2609_mult__eq__self__implies__10,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) )
     => ( ( Nb = one_one_nat )
        | ( Mb = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2610_mod__Suc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,Mb)),Nb) = $ite(aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = Nb,zero_zero_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))) ).

% mod_Suc
tff(fact_2611_mod__less__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2612_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_2613_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = $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_2614_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_2615_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_2616_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_2617_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_2618_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : aa_num_num(pow(Xb),one) = Xb ).

% pow.simps(1)
tff(fact_2619_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_2620_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_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_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_2628_mod__eq__0D,axiom,
    ! [Mb: nat,D: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),D) = zero_zero_nat )
     => ? [Q3: nat] : Mb = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,D),Q3) ) ).

% mod_eq_0D
tff(fact_2629_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2630_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_2631_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_2632_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_2633_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(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2634_unit__eq__div1,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B2) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_2635_unit__eq__div1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_2636_unit__eq__div1,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ).

% unit_eq_div1
tff(fact_2637_unit__eq__div2,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B2) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_2638_unit__eq__div2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B2) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_2639_unit__eq__div2,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B2) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) = C ) ) ) ).

% unit_eq_div2
tff(fact_2640_div__mult__unit2,axiom,
    ! [C: code_integer,B2: 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,B2),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B2),C) ) ) ) ).

% div_mult_unit2
tff(fact_2641_div__mult__unit2,axiom,
    ! [C: nat,B2: 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,B2),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2642_div__mult__unit2,axiom,
    ! [C: int,B2: 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,B2),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ) ).

% div_mult_unit2
tff(fact_2643_unit__div__commute,axiom,
    ! [B2: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B2)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B2) ) ) ).

% unit_div_commute
tff(fact_2644_unit__div__commute,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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),B2)),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)),B2) ) ) ).

% unit_div_commute
tff(fact_2645_unit__div__commute,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B2) ) ) ).

% unit_div_commute
tff(fact_2646_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B2: 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(B2,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2),C) ) ) ).

% unit_div_mult_swap
tff(fact_2647_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2648_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),C) ) ) ).

% unit_div_mult_swap
tff(fact_2649_is__unit__div__mult2__eq,axiom,
    ! [B2: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),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,B2),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B2),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2650_is__unit__div__mult2__eq,axiom,
    ! [B2: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2651_is__unit__div__mult2__eq,axiom,
    ! [B2: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),one_one_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2652_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,Mb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),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,Mb),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2653_mult__less__le__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2654_mult__less__le__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2655_mult__less__le__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2656_mult__less__le__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2657_mult__le__less__imp__less,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2658_mult__le__less__imp__less,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2659_mult__le__less__imp__less,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2660_mult__le__less__imp__less,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2661_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2662_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2663_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2664_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) ) ) ).

% mult_right_le_imp_le
tff(fact_2665_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B2: 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),B2))
     => ( 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),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2666_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
     => ( 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),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2667_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
     => ( 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),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2668_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
     => ( 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),B2) ) ) ).

% mult_left_le_imp_le
tff(fact_2669_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B2: 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),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2670_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2671_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2672_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B2: 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),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2673_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2674_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2675_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2676_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2677_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2678_mult__strict__mono_H,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2679_mult__strict__mono_H,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),D)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2680_mult__strict__mono_H,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2681_mult__strict__mono_H,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2682_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2683_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2684_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B2: 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,B2),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),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2685_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) ) ) ).

% mult_right_less_imp_less
tff(fact_2686_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2687_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2688_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2689_mult__strict__mono,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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),B2)
         => ( 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,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2690_mult__strict__mono,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( 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),B2)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2691_mult__strict__mono,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( 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),B2)
         => ( 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,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2692_mult__strict__mono,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( 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),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2693_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B2: 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),B2))
     => ( 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),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2694_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
     => ( 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),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2695_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B2: 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),B2))
     => ( 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),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2696_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
     => ( 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),B2) ) ) ).

% mult_left_less_imp_less
tff(fact_2697_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2698_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2699_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2700_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B2: 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),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2701_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2702_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2703_add__strict__increasing2,axiom,
    ! [A: real,B2: 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,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2704_add__strict__increasing2,axiom,
    ! [A: rat,B2: 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,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2705_add__strict__increasing2,axiom,
    ! [A: nat,B2: 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,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2706_add__strict__increasing2,axiom,
    ! [A: int,B2: 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,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2707_add__strict__increasing,axiom,
    ! [A: real,B2: 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,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2708_add__strict__increasing,axiom,
    ! [A: rat,B2: 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,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2709_add__strict__increasing,axiom,
    ! [A: nat,B2: 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,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2710_add__strict__increasing,axiom,
    ! [A: int,B2: 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,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2711_add__pos__nonneg,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2712_add__pos__nonneg,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2713_add__pos__nonneg,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2714_add__pos__nonneg,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_pos_nonneg
tff(fact_2715_add__nonpos__neg,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_nonpos_neg
tff(fact_2716_add__nonpos__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2717_add__nonpos__neg,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2718_add__nonpos__neg,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2719_add__nonneg__pos,axiom,
    ! [A: real,B2: 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),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2720_add__nonneg__pos,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2721_add__nonneg__pos,axiom,
    ! [A: nat,B2: 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),B2)
       => 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),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2722_add__nonneg__pos,axiom,
    ! [A: int,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% add_nonneg_pos
tff(fact_2723_add__neg__nonpos,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),zero_zero_real) ) ) ).

% add_neg_nonpos
tff(fact_2724_add__neg__nonpos,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2725_add__neg__nonpos,axiom,
    ! [A: nat,B2: 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,B2),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),B2)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2726_add__neg__nonpos,axiom,
    ! [A: int,B2: 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,B2),zero_zero_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2727_field__le__epsilon,axiom,
    ! [Xb: real,Y: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),E2)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2728_field__le__epsilon,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),E2)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_epsilon
tff(fact_2729_div__positive,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),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),B2)) ) ) ).

% div_positive
tff(fact_2730_div__positive,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% div_positive
tff(fact_2731_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B2: 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),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2732_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_2733_divide__nonpos__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonpos_pos
tff(fact_2734_divide__nonpos__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonpos_pos
tff(fact_2735_divide__nonpos__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2736_divide__nonpos__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonpos_neg
tff(fact_2737_divide__nonneg__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2738_divide__nonneg__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% divide_nonneg_pos
tff(fact_2739_divide__nonneg__neg,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),zero_zero_real) ) ) ).

% divide_nonneg_neg
tff(fact_2740_divide__nonneg__neg,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),zero_zero_rat) ) ) ).

% divide_nonneg_neg
tff(fact_2741_divide__le__cancel,axiom,
    ! [A: real,C: real,B2: 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,B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% divide_le_cancel
tff(fact_2742_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B2: 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(B2),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),B2) )
        & ( 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,B2),A) ) ) ) ).

% divide_le_cancel
tff(fact_2743_frac__less2,axiom,
    ! [Xb: real,Y: real,W: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W),Zb)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2744_frac__less2,axiom,
    ! [Xb: rat,Y: rat,W: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W),Zb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Zb)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less2
tff(fact_2745_frac__less,axiom,
    ! [Xb: real,Y: real,W: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Zb)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2746_frac__less,axiom,
    ! [Xb: rat,Y: rat,W: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Zb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Zb)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_less
tff(fact_2747_frac__le,axiom,
    ! [Y: real,Xb: real,W: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),W)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,W),Zb)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2748_frac__le,axiom,
    ! [Y: rat,Xb: rat,W: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),W)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,W),Zb)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Zb)),aa_rat_rat(divide_divide_rat(Y),W)) ) ) ) ) ).

% frac_le
tff(fact_2749_mult__left__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2750_mult__left__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2751_mult__left__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)),Xb) ) ) ) ).

% mult_left_le_one_le
tff(fact_2752_mult__right__le__one__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2753_mult__right__le__one__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2754_mult__right__le__one__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)),Xb) ) ) ) ).

% mult_right_le_one_le
tff(fact_2755_mult__le__one,axiom,
    ! [A: real,B2: 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),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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),B2)),one_one_real) ) ) ) ).

% mult_le_one
tff(fact_2756_mult__le__one,axiom,
    ! [A: rat,B2: 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),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),one_one_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_2757_mult__le__one,axiom,
    ! [A: nat,B2: 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),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),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),B2)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_2758_mult__le__one,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_2759_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_2760_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),one_one_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2761_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_2762_mult__left__le,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_2763_sum__squares__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2764_sum__squares__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2765_sum__squares__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2766_sum__squares__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2767_sum__squares__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2768_sum__squares__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2769_power__less__imp__less__base,axiom,
    ! [A: real,Nb: nat,B2: 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,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_2770_power__less__imp__less__base,axiom,
    ! [A: rat,Nb: nat,B2: 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(B2),Nb))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_2771_power__less__imp__less__base,axiom,
    ! [A: nat,Nb: nat,B2: 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,B2),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_2772_power__less__imp__less__base,axiom,
    ! [A: int,Nb: nat,B2: 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(B2),Nb))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% power_less_imp_less_base
tff(fact_2773_sum__squares__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2774_sum__squares__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2775_sum__squares__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2776_not__sum__squares__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y))),zero_zero_real) ).

% not_sum_squares_lt_zero
tff(fact_2777_not__sum__squares__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2778_not__sum__squares__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2779_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2780_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2781_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_2782_zero__less__two,axiom,
    aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2783_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_2784_zero__less__two,axiom,
    aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int)) ).

% zero_less_two
tff(fact_2785_divide__strict__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( 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),B2))
         => 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),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2786_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => 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),B2)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2787_divide__strict__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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),B2))
         => 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),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2788_divide__strict__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => 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),B2)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2789_mult__imp__less__div__pos,axiom,
    ! [Y: real,Zb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Zb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2790_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Zb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Zb),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2791_mult__imp__div__pos__less,axiom,
    ! [Y: real,Xb: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Zb) ) ) ).

% mult_imp_div_pos_less
tff(fact_2792_mult__imp__div__pos__less,axiom,
    ! [Y: rat,Xb: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Zb) ) ) ).

% mult_imp_div_pos_less
tff(fact_2793_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2794_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2) ) ) ).

% pos_less_divide_eq
tff(fact_2795_pos__divide__less__eq,axiom,
    ! [C: real,B2: 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,B2),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2796_pos__divide__less__eq,axiom,
    ! [C: rat,B2: 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(B2),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2797_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2798_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2799_neg__divide__less__eq,axiom,
    ! [C: real,B2: 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,B2),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2800_neg__divide__less__eq,axiom,
    ! [C: rat,B2: 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(B2),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2) ) ) ).

% neg_divide_less_eq
tff(fact_2801_less__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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)),B2),
          $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,B2),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_2802_less__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2803_divide__less__eq,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% divide_less_eq
tff(fact_2804_divide__less__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2805_less__divide__eq__1,axiom,
    ! [B2: 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,B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2806_less__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2807_divide__less__eq__1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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,B2),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),B2) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
tff(fact_2808_divide__less__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),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),B2) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2809_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_2810_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_2811_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_2812_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_2813_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = numera6690914467698888265omplex(W) )
    <=> $ite(C != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C),numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2814_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) = aa_num_real(numeral_numeral_real,W) )
    <=> $ite(C != zero_zero_real,B2 = 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_2815_divide__eq__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C) = aa_num_rat(numeral_numeral_rat,W) )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_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_2816_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( numera6690914467698888265omplex(W) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),C) = B2,numera6690914467698888265omplex(W) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2817_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: real,C: real] :
      ( ( aa_num_real(numeral_numeral_real,W) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),C) = B2,aa_num_real(numeral_numeral_real,W) = zero_zero_real) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2818_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( aa_num_rat(numeral_numeral_rat,W) = aa_rat_rat(divide_divide_rat(B2),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C) = B2,aa_num_rat(numeral_numeral_rat,W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2819_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: complex,Zb: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Zb)),B2) = $ite(Zb = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),aa_complex_complex(times_times_complex(B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(2)
tff(fact_2820_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: real,Zb: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Zb)),B2) = $ite(Zb = zero_zero_real,B2,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,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(2)
tff(fact_2821_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Zb: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(A),Zb)),B2) = $ite(Zb = zero_zero_rat,B2,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(2)
tff(fact_2822_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: complex,B2: complex,Zb: complex] :
      aa_complex_complex(plus_plus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Zb)) = $ite(Zb = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(A),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(1)
tff(fact_2823_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: real,B2: real,Zb: real] :
      aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Zb)) = $ite(Zb = 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),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(1)
tff(fact_2824_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B2: rat,Zb: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(divide_divide_rat(B2),Zb)) = $ite(Zb = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(1)
tff(fact_2825_add__frac__eq,axiom,
    ! [Y: complex,Zb: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Zb != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Zb)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Zb)) ) ) ) ).

% add_frac_eq
tff(fact_2826_add__frac__eq,axiom,
    ! [Y: real,Zb: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Zb != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb)) ) ) ) ).

% add_frac_eq
tff(fact_2827_add__frac__eq,axiom,
    ! [Y: rat,Zb: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Zb != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Zb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb)) ) ) ) ).

% add_frac_eq
tff(fact_2828_add__frac__num,axiom,
    ! [Y: complex,Xb: complex,Zb: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),Zb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Zb),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2829_add__frac__num,axiom,
    ! [Y: real,Xb: real,Zb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Zb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2830_add__frac__num,axiom,
    ! [Y: rat,Xb: rat,Zb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Zb) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2831_add__num__frac,axiom,
    ! [Y: complex,Zb: complex,Xb: complex] :
      ( ( Y != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Zb),aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Zb),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2832_add__num__frac,axiom,
    ! [Y: real,Zb: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2833_add__num__frac,axiom,
    ! [Y: rat,Zb: rat,Xb: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_rat_rat(divide_divide_rat(Xb),Y)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2834_add__divide__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Xb),Zb)),Y)),Zb) ) ) ).

% add_divide_eq_iff
tff(fact_2835_add__divide__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),Y)),Zb) ) ) ).

% add_divide_eq_iff
tff(fact_2836_add__divide__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(divide_divide_rat(Y),Zb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),Y)),Zb) ) ) ).

% add_divide_eq_iff
tff(fact_2837_divide__add__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Zb)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(times_times_complex(Y),Zb))),Zb) ) ) ).

% divide_add_eq_iff
tff(fact_2838_divide__add__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),Zb) ) ) ).

% divide_add_eq_iff
tff(fact_2839_divide__add__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(divide_divide_rat(Xb),Zb)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),Zb) ) ) ).

% divide_add_eq_iff
tff(fact_2840_power__le__imp__le__base,axiom,
    ! [A: real,Nb: nat,B2: 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,B2),aa_nat_nat(suc,Nb)))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_2841_power__le__imp__le__base,axiom,
    ! [A: rat,Nb: nat,B2: 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(B2),aa_nat_nat(suc,Nb)))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_2842_power__le__imp__le__base,axiom,
    ! [A: nat,Nb: nat,B2: 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,B2),aa_nat_nat(suc,Nb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_2843_power__le__imp__le__base,axiom,
    ! [A: int,Nb: nat,B2: 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(B2),aa_nat_nat(suc,Nb)))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% power_le_imp_le_base
tff(fact_2844_power__inject__base,axiom,
    ! [A: real,Nb: nat,B2: 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,B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2845_power__inject__base,axiom,
    ! [A: rat,Nb: nat,B2: rat] :
      ( ( aa_nat_rat(power_power_rat(A),aa_nat_nat(suc,Nb)) = aa_nat_rat(power_power_rat(B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2846_power__inject__base,axiom,
    ! [A: nat,Nb: nat,B2: 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,B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2847_power__inject__base,axiom,
    ! [A: int,Nb: nat,B2: int] :
      ( ( aa_nat_int(power_power_int(A),aa_nat_nat(suc,Nb)) = aa_nat_int(power_power_int(B2),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),B2)
         => ( A = B2 ) ) ) ) ).

% power_inject_base
tff(fact_2848_div__add__self1,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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,B2),A)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2849_div__add__self1,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2850_div__add__self2,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2851_div__add__self2,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2852_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: complex,B2: complex,Zb: complex] :
      aa_complex_complex(minus_minus_complex(A),aa_complex_complex(divide1717551699836669952omplex(B2),Zb)) = $ite(Zb = zero_zero_complex,A,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(A),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(4)
tff(fact_2853_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: real,B2: real,Zb: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),Zb)) = $ite(Zb = 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),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(4)
tff(fact_2854_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B2: rat,Zb: rat] :
      aa_rat_rat(minus_minus_rat(A),aa_rat_rat(divide_divide_rat(B2),Zb)) = $ite(Zb = zero_zero_rat,A,aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Zb)),B2)),Zb)) ).

% add_divide_eq_if_simps(4)
tff(fact_2855_diff__frac__eq,axiom,
    ! [Y: complex,Zb: complex,Xb: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Zb != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Y)),aa_complex_complex(divide1717551699836669952omplex(W),Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Zb)),aa_complex_complex(times_times_complex(W),Y))),aa_complex_complex(times_times_complex(Y),Zb)) ) ) ) ).

% diff_frac_eq
tff(fact_2856_diff__frac__eq,axiom,
    ! [Y: real,Zb: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Zb != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb)) ) ) ) ).

% diff_frac_eq
tff(fact_2857_diff__frac__eq,axiom,
    ! [Y: rat,Zb: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Zb != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Zb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb)) ) ) ) ).

% diff_frac_eq
tff(fact_2858_diff__divide__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(Y),Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(Xb),Zb)),Y)),Zb) ) ) ).

% diff_divide_eq_iff
tff(fact_2859_diff__divide__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Y),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),Y)),Zb) ) ) ).

% diff_divide_eq_iff
tff(fact_2860_diff__divide__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(Y),Zb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),Y)),Zb) ) ) ).

% diff_divide_eq_iff
tff(fact_2861_divide__diff__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(Xb),Zb)),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Xb),aa_complex_complex(times_times_complex(Y),Zb))),Zb) ) ) ).

% divide_diff_eq_iff
tff(fact_2862_divide__diff__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),Zb) ) ) ).

% divide_diff_eq_iff
tff(fact_2863_divide__diff__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(Xb),Zb)),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),Zb) ) ) ).

% divide_diff_eq_iff
tff(fact_2864_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),modulo364778990260209775nteger(A,B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2865_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => 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),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2866_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2867_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2)
       => ( modulo364778990260209775nteger(A,B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2868_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B2: 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),B2)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2869_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2870_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_2871_cong__exp__iff__simps_I2_J,axiom,
    ! [Nb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_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(aa_int_fun_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_2872_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_2873_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_2874_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_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_2875_cong__exp__iff__simps_I1_J,axiom,
    ! [Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(one)) = zero_z3403309356797280102nteger ).

% cong_exp_iff_simps(1)
tff(fact_2876_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_2877_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)
            & ! [I4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I4),K2)
               => ~ aa_nat_o(P,I4) )
            & aa_nat_o(P,aa_nat_nat(suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_2878_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_2879_n__less__n__mult__m,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Mb)) ) ) ).

% n_less_n_mult_m
tff(fact_2880_n__less__m__mult__n,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_2881_one__less__mult,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ) ) ).

% one_less_mult
tff(fact_2882_realpow__pos__nth2,axiom,
    ! [A: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ? [R3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
          & ( aa_nat_real(aa_real_fun_nat_real(power_power_real,R3),aa_nat_nat(suc,Nb)) = A ) ) ) ).

% realpow_pos_nth2
tff(fact_2883_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)
       => ( ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(P,N2)
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_2884_nat__mult__le__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),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,Mb),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_2885_div__greater__zero__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_2886_div__le__mono2,axiom,
    ! [Mb: nat,Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,K),Mb)) ) ) ).

% div_le_mono2
tff(fact_2887_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_2888_nat__diff__split,axiom,
    ! [P: fun_nat_o,A: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
         => aa_nat_o(P,zero_zero_nat) )
        & ! [D3: nat] :
            ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
           => aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split
tff(fact_2889_nat__diff__split__asm,axiom,
    ! [P: fun_nat_o,A: nat,B2: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2))
    <=> ~ ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
            & ~ aa_nat_o(P,zero_zero_nat) )
          | ? [D3: nat] :
              ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),D3) )
              & ~ aa_nat_o(P,D3) ) ) ) ).

% nat_diff_split_asm
tff(fact_2890_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),I)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb)) ) ).

% nat_one_le_power
tff(fact_2891_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),N2)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_2892_div__less__iff__less__mult,axiom,
    ! [Q2: nat,Mb: 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,Mb),Q2)),Nb)
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_2893_nat__mult__div__cancel1,axiom,
    ! [K: nat,Mb: 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),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_2894_div__eq__dividend__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = Mb )
      <=> ( Nb = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2895_div__less__dividend,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Mb) ) ) ).

% div_less_dividend
tff(fact_2896_mod__le__divisor,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2897_div__less__mono,axiom,
    ! [A3: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( 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,A3),Nb) = zero_zero_nat )
         => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),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,A3),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2898_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_2899_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_2900_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(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4) ) ).

% evenE
tff(fact_2901_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_2902_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_2903_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_2904_odd__even__add,axiom,
    ! [A: code_integer,B2: 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))),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),plus_p5714425477246183910nteger(A,B2)) ) ) ).

% odd_even_add
tff(fact_2905_odd__even__add,axiom,
    ! [A: nat,B2: 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))),B2)
       => 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),B2)) ) ) ).

% odd_even_add
tff(fact_2906_odd__even__add,axiom,
    ! [A: int,B2: 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))),B2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ) ) ).

% odd_even_add
tff(fact_2907_bit__eq__rec,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = B2 )
    <=> ( ( 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))),B2) )
        & ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = divide6298287555418463151nteger(B2,numera6620942414471956472nteger(aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2908_bit__eq__rec,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( ( 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))),B2) )
        & ( 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,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2909_bit__eq__rec,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( ( 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))),B2) )
        & ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ) ).

% bit_eq_rec
tff(fact_2910_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_2911_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_2912_even__mask__div__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% even_mask_div_iff
tff(fact_2913_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R2: nat,Mb: 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),Mb))
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Q2))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),Mb)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_2914_power__dvd__imp__le,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_2915_mod__nat__eqI,axiom,
    ! [R2: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,R2),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,R2),Mb)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),R2))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2916_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_2917_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2918_mult__less__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2919_mult__less__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),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),B2) )
        & ( 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,B2),one_one_real) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2920_mult__less__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) )
        & ( 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,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2921_mult__less__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) )
        & ( 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,B2),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2922_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_2923_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2924_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2925_mult__less__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_real) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2926_mult__less__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2927_mult__less__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2928_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_2929_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2930_mult__le__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2931_mult__le__cancel__right1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),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),B2) )
        & ( 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,B2),one_one_real) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2932_mult__le__cancel__right1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),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),B2) )
        & ( 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,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2933_mult__le__cancel__right1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2) )
        & ( 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,B2),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2934_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_2935_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2936_mult__le__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),C)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),C)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),zero_zero_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2937_mult__le__cancel__left1,axiom,
    ! [C: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_real) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2938_mult__le__cancel__left1,axiom,
    ! [C: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2939_mult__le__cancel__left1,axiom,
    ! [C: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))
    <=> ( ( 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),B2) )
        & ( 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,B2),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_2940_field__le__mult__one__interval,axiom,
    ! [Xb: real,Y: real] :
      ( ! [Z: 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,Z),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Z),Xb)),Y) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2941_field__le__mult__one__interval,axiom,
    ! [Xb: rat,Y: rat] :
      ( ! [Z: 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,Z),one_one_rat)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Xb)),Y) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% field_le_mult_one_interval
tff(fact_2942_divide__left__mono__neg,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( 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),B2))
         => 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),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2943_divide__left__mono__neg,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => 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),B2)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2944_mult__imp__le__div__pos,axiom,
    ! [Y: real,Zb: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y)),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Zb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2945_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Zb: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y)),Xb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Zb),aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2946_mult__imp__div__pos__le,axiom,
    ! [Y: real,Xb: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),Zb) ) ) ).

% mult_imp_div_pos_le
tff(fact_2947_mult__imp__div__pos__le,axiom,
    ! [Y: rat,Xb: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y))
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),Zb) ) ) ).

% mult_imp_div_pos_le
tff(fact_2948_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2949_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2) ) ) ).

% pos_le_divide_eq
tff(fact_2950_pos__divide__le__eq,axiom,
    ! [C: real,B2: 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,B2),C)),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2951_pos__divide__le__eq,axiom,
    ! [C: rat,B2: 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(B2),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2952_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2953_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2954_neg__divide__le__eq,axiom,
    ! [C: real,B2: 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,B2),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)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2955_neg__divide__le__eq,axiom,
    ! [C: rat,B2: 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(B2),C)),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2) ) ) ).

% neg_divide_le_eq
tff(fact_2956_divide__left__mono,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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),B2))
         => 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),B2)) ) ) ) ).

% divide_left_mono
tff(fact_2957_divide__left__mono,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),C)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => 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),B2)) ) ) ) ).

% divide_left_mono
tff(fact_2958_le__divide__eq,axiom,
    ! [A: real,B2: 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,B2),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)),B2),
          $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,B2),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_2959_le__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2960_divide__le__eq,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% divide_le_eq
tff(fact_2961_divide__le__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B2),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2962_le__divide__eq__1,axiom,
    ! [B2: 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,B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2963_le__divide__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(divide_divide_rat(B2),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),B2) )
        | ( 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,B2),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2964_divide__le__eq__1,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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,B2),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),B2) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
tff(fact_2965_divide__le__eq__1,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),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),B2) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2966_convex__bound__le,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2967_convex__bound__le,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2968_convex__bound__le,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2969_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),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_2970_divide__less__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B2),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_2971_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: 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,B2),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)),B2),
          $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,B2),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_2972_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: 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(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_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_2973_frac__le__eq,axiom,
    ! [Y: real,Zb: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Zb != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Zb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),zero_zero_real) ) ) ) ).

% frac_le_eq
tff(fact_2974_frac__le__eq,axiom,
    ! [Y: rat,Zb: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Zb != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Zb))
        <=> 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(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_2975_frac__less__eq,axiom,
    ! [Y: real,Zb: real,Xb: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Zb != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,W),Zb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),zero_zero_real) ) ) ) ).

% frac_less_eq
tff(fact_2976_frac__less__eq,axiom,
    ! [Y: rat,Zb: rat,Xb: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Zb != zero_zero_rat )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(Xb),Y)),aa_rat_rat(divide_divide_rat(W),Zb))
        <=> 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(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_2977_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_2978_power__Suc__less,axiom,
    ! [A: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(power_power_rat(A),Nb))),aa_nat_rat(power_power_rat(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2979_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_2980_power__Suc__less,axiom,
    ! [A: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),Nb))),aa_nat_int(power_power_int(A),Nb)) ) ) ).

% power_Suc_less
tff(fact_2981_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_2982_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_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2990_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2991_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2992_power__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
     => ( 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),N)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_strict_decreasing
tff(fact_2993_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2994_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2995_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2996_power__decreasing,axiom,
    ! [Nb: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
     => ( 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),N)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_decreasing
tff(fact_2997_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_2998_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_2999_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_3000_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_3001_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_3002_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_3003_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_3004_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_3005_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_3006_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_3007_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_3008_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_3009_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_3010_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_3011_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_3012_power__diff,axiom,
    ! [A: complex,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_complex(power_power_complex(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(power_power_complex(A),Mb)),aa_nat_complex(power_power_complex(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3013_power__diff,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3014_power__diff,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(power_power_rat(A),Mb)),aa_nat_rat(power_power_rat(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3015_power__diff,axiom,
    ! [A: nat,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) ) ) ) ).

% power_diff
tff(fact_3016_power__diff,axiom,
    ! [A: int,Nb: nat,Mb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) ) ) ) ).

% power_diff
tff(fact_3017_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb)),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3018_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3019_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        | ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3020_div__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ).

% div_if
tff(fact_3021_div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_3022_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_3023_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3024_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,Mb: 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,Mb),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,Mb),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_3025_add__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,Nb,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% add_eq_if
tff(fact_3026_dividend__less__times__div,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)))) ) ).

% dividend_less_times_div
tff(fact_3027_dividend__less__div__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_3028_split__div,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,zero_zero_nat) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J3) )
               => aa_nat_o(P,I2) ) ) ) ) ) ).

% split_div
tff(fact_3029_mult__eq__if,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = $ite(Mb = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)),Nb))) ).

% mult_eq_if
tff(fact_3030_split__mod,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_nat_o(P,Mb) )
        & ( ( Nb != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J3),Nb)
             => ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),I2)),J3) )
               => aa_nat_o(P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_3031_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_3032_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_3033_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_3034_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_3035_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(aa_int_fun_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_3036_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_3037_power__mono__odd,axiom,
    ! [Nb: nat,A: real,B2: 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),B2)
       => 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,B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_3038_power__mono__odd,axiom,
    ! [Nb: nat,A: rat,B2: 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),B2)
       => 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(B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_3039_power__mono__odd,axiom,
    ! [Nb: nat,A: int,B2: 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),B2)
       => 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(B2),Nb)) ) ) ).

% power_mono_odd
tff(fact_3040_dvd__power__iff__le,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,K),Mb)),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,Mb),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_3041_convex__bound__lt,axiom,
    ! [Xb: real,A: real,Y: real,U: real,V: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),A)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),U)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),V)
           => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,U),V) = one_one_real )
             => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3042_convex__bound__lt,axiom,
    ! [Xb: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),A)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),U)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),V)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V) = one_one_rat )
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3043_convex__bound__lt,axiom,
    ! [Xb: int,A: int,Y: int,U: int,V: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),U)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),V)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V) = one_one_int )
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_3044_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: 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,B2),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)),B2),
          $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,B2),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_3045_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B2: 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(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_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_3046_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),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_3047_divide__le__eq__numeral_I1_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),C)),B2),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_3048_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_3049_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_3050_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_3051_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_3052_scaling__mono,axiom,
    ! [U: real,V: real,R2: real,S: 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),S)
         => 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))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3053_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: 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),S)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(minus_minus_rat(V),U))),S))),V) ) ) ) ).

% scaling_mono
tff(fact_3054_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_3055_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_3056_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_3057_power2__eq__imp__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3058_power2__eq__imp__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3059_power2__eq__imp__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3060_power2__eq__imp__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
         => ( Xb = Y ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_3061_power2__le__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3062_power2__le__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3063_power2__le__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3064_power2__le__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% power2_le_imp_le
tff(fact_3065_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_3066_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_3067_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_3068_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),C)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B2),C)),modulo364778990260209775nteger(A,B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3069_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B2: 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,B2),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3070_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),C)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_3071_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3072_exp__add__not__zero__imp__left,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
tff(fact_3073_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3074_exp__add__not__zero__imp__right,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
tff(fact_3075_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3076_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) != zero_zero_int )
     => ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_3077_power__diff__power__eq,axiom,
    ! [A: nat,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3078_power__diff__power__eq,axiom,
    ! [A: int,Mb: nat,Nb: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),Mb)),aa_nat_int(power_power_int(A),Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)))) ) ) ).

% power_diff_power_eq
tff(fact_3079_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_3080_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_3081_nat__induct2,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( aa_nat_o(P,one_one_nat)
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_induct2
tff(fact_3082_power__eq__if,axiom,
    ! [P2: complex,Mb: nat] :
      aa_nat_complex(power_power_complex(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(P2),aa_nat_complex(power_power_complex(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3083_power__eq__if,axiom,
    ! [P2: real,Mb: nat] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,P2),aa_nat_real(aa_real_fun_nat_real(power_power_real,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3084_power__eq__if,axiom,
    ! [P2: rat,Mb: nat] :
      aa_nat_rat(power_power_rat(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),aa_nat_rat(power_power_rat(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3085_power__eq__if,axiom,
    ! [P2: nat,Mb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3086_power__eq__if,axiom,
    ! [P2: int,Mb: nat] :
      aa_nat_int(power_power_int(P2),Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P2),aa_nat_int(power_power_int(P2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% power_eq_if
tff(fact_3087_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_3088_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_3089_power__minus__mult,axiom,
    ! [Nb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_rat(power_power_rat(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3090_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_3091_power__minus__mult,axiom,
    ! [Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),A) = aa_nat_int(power_power_int(A),Nb) ) ) ).

% power_minus_mult
tff(fact_3092_le__div__geq,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_3093_split__div_H,axiom,
    ! [P: fun_nat_o,Mb: nat,Nb: nat] :
      ( aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb))
    <=> ( ( ( Nb = zero_zero_nat )
          & aa_nat_o(P,zero_zero_nat) )
        | ? [Q4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q4)),Mb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(suc,Q4)))
            & aa_nat_o(P,Q4) ) ) ) ).

% split_div'
tff(fact_3094_Suc__times__mod__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb))),Mb) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_3095_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_3096_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_3097_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B4)),one_one_int) ) ).

% oddE
tff(fact_3098_power2__less__imp__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3099_power2__less__imp__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3100_power2__less__imp__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3101_power2__less__imp__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% power2_less_imp_less
tff(fact_3102_sum__power2__ge__zero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3103_sum__power2__ge__zero,axiom,
    ! [Xb: rat,Y: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3104_sum__power2__ge__zero,axiom,
    ! [Xb: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% sum_power2_ge_zero
tff(fact_3105_sum__power2__le__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real)
    <=> ( ( Xb = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3106_sum__power2__le__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat)
    <=> ( ( Xb = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3107_sum__power2__le__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int)
    <=> ( ( Xb = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_3108_not__sum__power2__lt__zero,axiom,
    ! [Xb: real,Y: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_real) ).

% not_sum_power2_lt_zero
tff(fact_3109_not__sum__power2__lt__zero,axiom,
    ! [Xb: rat,Y: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_rat) ).

% not_sum_power2_lt_zero
tff(fact_3110_not__sum__power2__lt__zero,axiom,
    ! [Xb: int,Y: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),zero_zero_int) ).

% not_sum_power2_lt_zero
tff(fact_3111_sum__power2__gt__zero__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3112_sum__power2__gt__zero__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3113_sum__power2__gt__zero__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
    <=> ( ( Xb != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_3114_divmod__digit__0_I2_J,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( 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))),B2))),B2)
       => ( 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))),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3115_divmod__digit__0_I2_J,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3116_divmod__digit__0_I2_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)) = modulo364778990260209775nteger(A,B2) ) ) ) ).

% divmod_digit_0(2)
tff(fact_3117_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_3118_bits__stable__imp__add__self,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = zero_zero_int ) ) ).

% bits_stable_imp_add_self
tff(fact_3119_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_3120_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_3121_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_3122_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_3123_nat__bit__induct,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ! [N2: nat] :
            ( aa_nat_o(P,N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => 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))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa_nat_o(P,N2)
             => 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))),N2))) )
         => aa_nat_o(P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_3124_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_3125_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_3126_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A3),B3)
     => ( 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,A3),Nb)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B3),Nb) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B3),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_3127_divmod__digit__0_I1_J,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B2)
     => ( 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))),B2))),B2)
       => ( 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))),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3128_divmod__digit__0_I1_J,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3129_divmod__digit__0_I1_J,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),B2)
       => ( 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))),B2))) = divide6298287555418463151nteger(A,B2) ) ) ) ).

% divmod_digit_0(1)
tff(fact_3130_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_3131_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_3132_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_3133_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_3134_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_3135_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_3136_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3137_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3138_even__mask__div__iff_H,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),one_one_int)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% even_mask_div_iff'
tff(fact_3139_mod__double__modulus,axiom,
    ! [Mb: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Mb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
       => ( ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = modulo364778990260209775nteger(Xb,Mb) )
          | ( modulo364778990260209775nteger(Xb,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Mb)) = plus_p5714425477246183910nteger(modulo364778990260209775nteger(Xb,Mb),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3140_mod__double__modulus,axiom,
    ! [Mb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),Xb)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3141_mod__double__modulus,axiom,
    ! [Mb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Mb)),Mb) ) ) ) ) ).

% mod_double_modulus
tff(fact_3142_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B2: 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),B2)
       => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)),B2) = modulo364778990260209775nteger(A,B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3143_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B2: 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),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),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))),B2)))
         => ( 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))),B2))),B2) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3144_divmod__digit__1_I2_J,axiom,
    ! [A: int,B2: 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),B2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_3145_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_3146_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: nat,B2: 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,B2),Nb))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_3147_pow__divides__pow__iff,axiom,
    ! [Nb: nat,A: int,B2: 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(B2),Nb))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ) ).

% pow_divides_pow_iff
tff(fact_3148_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_3149_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_3150_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_3151_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_3152_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
tff(fact_3153_double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% double_eq_0_iff
tff(fact_3154_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)
     => ! [X4: code_integer,K4: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,Ta))
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(3)
tff(fact_3155_inf__period_I3_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X4: real,K4: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),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,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3156_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X4: rat,K4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),Ta))
        <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3157_inf__period_I3_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int,K4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(3)
tff(fact_3158_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)
     => ! [X4: code_integer,K4: code_integer] :
          ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,Ta))
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(minus_8373710615458151222nteger(X4,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K4),D4)),Ta)) ) ) ).

% inf_period(4)
tff(fact_3159_inf__period_I4_J,axiom,
    ! [D: real,D4: real,Ta: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),D4)
     => ! [X4: real,K4: real] :
          ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),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,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3160_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,Ta: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),D4)
     => ! [X4: rat,K4: rat] :
          ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),Ta))
        <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3161_inf__period_I4_J,axiom,
    ! [D: int,D4: int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int,K4: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))),Ta)) ) ) ).

% inf_period(4)
tff(fact_3162_unity__coeff__ex,axiom,
    ! [P: fun_Code_integer_o,L: code_integer] :
      ( ? [X: code_integer] : aa_Code_integer_o(P,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X))
    <=> ? [X: code_integer] :
          ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),plus_p5714425477246183910nteger(X,zero_z3403309356797280102nteger))
          & aa_Code_integer_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_3163_unity__coeff__ex,axiom,
    ! [P: fun_real_o,L: real] :
      ( ? [X: real] : aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,L),X))
    <=> ? [X: real] :
          ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X),zero_zero_real))
          & aa_real_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_3164_unity__coeff__ex,axiom,
    ! [P: fun_rat_o,L: rat] :
      ( ? [X: rat] : aa_rat_o(P,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X))
    <=> ? [X: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),zero_zero_rat))
          & aa_rat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_3165_unity__coeff__ex,axiom,
    ! [P: fun_nat_o,L: nat] :
      ( ? [X: nat] : aa_nat_o(P,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X))
    <=> ? [X: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),zero_zero_nat))
          & aa_nat_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_3166_unity__coeff__ex,axiom,
    ! [P: fun_int_o,L: int] :
      ( ? [X: int] : aa_int_o(P,aa_int_int(aa_int_fun_int_int(times_times_int,L),X))
    <=> ? [X: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),zero_zero_int))
          & aa_int_o(P,X) ) ) ).

% unity_coeff_ex
tff(fact_3167_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(aa_nat_fun_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_3168_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_3169_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(aa_nat_fun_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_3170_set__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,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_3171_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_3172_unset__bit__negative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,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_3173_zle__add1__eq__le,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb) ) ).

% zle_add1_eq_le
tff(fact_3174_zle__diff1__eq,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Zb),one_one_int))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ).

% zle_diff1_eq
tff(fact_3175_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),L)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_3176_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_3177_le__imp__0__less,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Zb)) ) ).

% le_imp_0_less
tff(fact_3178_split__zmod,axiom,
    ! [P: fun_int_o,Nb: int,K: int] :
      ( aa_int_o(P,aa_int_int(aa_int_fun_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)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_int,J3),K)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
         => ! [I2: int,J3: int] :
              ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),J3)
                & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J3),zero_zero_int)
                & ( Nb = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I2)),J3) ) )
             => aa_int_o(P,J3) ) ) ) ) ).

% split_zmod
tff(fact_3179_odd__less__0__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Zb)),Zb)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),zero_zero_int) ) ).

% odd_less_0_iff
tff(fact_3180_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),K)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_3181_q__pos__lemma,axiom,
    ! [B: int,Q5: int,R4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),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,zero_zero_int),B)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_3182_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),I)),aa_int_int(aa_int_fun_int_int(times_times_int,K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_3183_int__mod__neg__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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,B2),R2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_3184_int__mod__pos__eq,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_3185_pos__zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
      <=> ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_3186_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(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z2: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P3,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P3,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_3187_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R4),B)
         => ( 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),B)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_3188_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(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [Z2: int] :
            ! [X3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
             => ( aa_int_o(P,X3)
              <=> aa_int_o(P1,X3) ) )
         => ( ? [X_12: int] : aa_int_o(P1,X_12)
           => ? [X_1: int] : aa_int_o(P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_3189_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(aa_int_fun_int_int(minus_minus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_3190_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(aa_int_fun_int_int(plus_plus_int,X3),D)) )
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K)
         => ! [X4: int] :
              ( aa_int_o(P,X4)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_3191_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R2: int,B: int,Q5: int,R4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q2)),R2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),Q5)),R4)),zero_zero_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B2)
         => ( 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),B)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),B2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_3192_int__one__le__iff__zero__less,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Zb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb) ) ).

% int_one_le_iff_zero_less
tff(fact_3193_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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),B2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,R2),B2)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_3194_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),Q5)),R4)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B2),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,B2),R2)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),R4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_3195_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_3196_zdvd__reduce,axiom,
    ! [K: int,Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),Nb) ) ).

% zdvd_reduce
tff(fact_3197_zdvd__period,axiom,
    ! [A: int,D: int,Xb: int,Ta: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Ta))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,C),D))),Ta)) ) ) ).

% zdvd_period
tff(fact_3198_odd__nonzero,axiom,
    ! [Zb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Zb)),Zb) != zero_zero_int ).

% odd_nonzero
tff(fact_3199_times__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),L) = zero_zero_int ).

% times_int_code(2)
tff(fact_3200_times__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(times_times_int,K),zero_zero_int) = zero_zero_int ).

% times_int_code(1)
tff(fact_3201_zmod__eq__0D,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
     => ? [Q3: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q3) ) ).

% zmod_eq_0D
tff(fact_3202_zdvd__mult__cancel,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb))
     => ( ( K != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3203_zdvd__mono,axiom,
    ! [K: int,Mb: int,Ta: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),Ta)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Ta)) ) ) ).

% zdvd_mono
tff(fact_3204_zmod__eq__0__iff,axiom,
    ! [Mb: int,D: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),D) = zero_zero_int )
    <=> ? [Q4: int] : Mb = aa_int_int(aa_int_fun_int_int(times_times_int,D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_3205_imult__is__0,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( times_7803423173614009249d_enat(Mb,Nb) = zero_z5237406670263579293d_enat )
    <=> ( ( Mb = zero_z5237406670263579293d_enat )
        | ( Nb = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
tff(fact_3206_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% signed_take_bit_mult
tff(fact_3207_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_3208_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% neg_mod_bound
tff(fact_3209_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_3210_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,L),zero_zero_int)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)),zero_zero_int) ) ).

% neg_mod_sign
tff(fact_3211_zmod__le__nonneg__dividend,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Mb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Mb),K)),Mb) ) ).

% zmod_le_nonneg_dividend
tff(fact_3212_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,I),K) = I )
    <=> ( ( K = zero_zero_int )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,I),K) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_3213_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
        | ( ( L = zero_zero_int )
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K) )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),L) ) ) ).

% mod_int_pos_iff
tff(fact_3214_pos__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_3215_neg__mod__conj,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),zero_zero_int)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ) ) ).

% neg_mod_conj
tff(fact_3216_zdvd__not__zless,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Mb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Mb),Nb)
       => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Nb),Mb) ) ) ).

% zdvd_not_zless
tff(fact_3217_zdvd__imp__le,axiom,
    ! [Zb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Zb),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,Zb),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3218_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_3219_plus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),L) = L ).

% plus_int_code(2)
tff(fact_3220_plus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,K),zero_zero_int) = K ).

% plus_int_code(1)
tff(fact_3221_minus__int__code_I1_J,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,K),zero_zero_int) = K ).

% minus_int_code(1)
tff(fact_3222_zdvd__zdiffD,axiom,
    ! [K: int,Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),Nb))
     => ( 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),Mb) ) ) ).

% zdvd_zdiffD
tff(fact_3223_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)) ).

% signed_take_bit_add
tff(fact_3224_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri631733984087533419it_int(Nb),K)),aa_int_int(bit_ri631733984087533419it_int(Nb),L))) = aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% signed_take_bit_diff
tff(fact_3225_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
tff(fact_3226_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(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K)) ).

% set_bit_greater_eq
tff(fact_3227_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K)),K) ).

% unset_bit_less_eq
tff(fact_3228_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(4)
tff(fact_3229_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(3)
tff(fact_3230_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa_int_int(aa_int_fun_int_int(times_times_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z1)),aa_int_int(aa_int_fun_int_int(times_times_int,W),Z22)) ).

% int_distrib(2)
tff(fact_3231_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z1),Z22)),W) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Z1),W)),aa_int_int(aa_int_fun_int_int(times_times_int,Z22),W)) ).

% int_distrib(1)
tff(fact_3232_zless__imp__add1__zle,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Zb) ) ).

% zless_imp_add1_zle
tff(fact_3233_int__less__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),K)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_less_induct
tff(fact_3234_zless__add1__eq,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),one_one_int))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb)
        | ( W = Zb ) ) ) ).

% zless_add1_eq
tff(fact_3235_int__le__induct,axiom,
    ! [I: int,K: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),K)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_le_induct
tff(fact_3236_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I)
     => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_gr_induct
tff(fact_3237_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I)
     => ( aa_int_o(P,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_ge_induct
tff(fact_3238_add1__zle__eq,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),one_one_int)),Zb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ).

% add1_zle_eq
tff(fact_3239_int__induct,axiom,
    ! [P: fun_int_o,K: int,I: int] :
      ( aa_int_o(P,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(P,I3)
             => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(P,I3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(P,I) ) ) ) ).

% int_induct
tff(fact_3240_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_diff_iff
tff(fact_3241_minf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X4) ) ).

% minf(7)
tff(fact_3242_minf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X4) ) ).

% minf(7)
tff(fact_3243_minf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X4) ) ).

% minf(7)
tff(fact_3244_minf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X4) ) ).

% minf(7)
tff(fact_3245_minf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4) ) ).

% minf(7)
tff(fact_3246_minf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Ta) ) ).

% minf(5)
tff(fact_3247_minf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Ta) ) ).

% minf(5)
tff(fact_3248_minf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Ta) ) ).

% minf(5)
tff(fact_3249_minf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Ta) ) ).

% minf(5)
tff(fact_3250_minf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta) ) ).

% minf(5)
tff(fact_3251_minf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_3252_minf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_3253_minf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_3254_minf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_3255_minf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(4)
tff(fact_3256_minf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_3257_minf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_3258_minf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_3259_minf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_3260_minf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ( X4 != Ta ) ) ).

% minf(3)
tff(fact_3261_minf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z2: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P3,X4)
                | aa_real_o(Q6,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_3262_minf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z2: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P3,X4)
                | aa_rat_o(Q6,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_3263_minf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z2: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P3,X4)
                | aa_num_o(Q6,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_3264_minf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z2: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P3,X4)
                | aa_nat_o(Q6,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_3265_minf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z2: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P3,X4)
                | aa_int_o(Q6,X4) ) ) ) ) ) ).

% minf(2)
tff(fact_3266_minf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z2: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Z2)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P3,X4)
                & aa_real_o(Q6,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_3267_minf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z2: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Z2)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P3,X4)
                & aa_rat_o(Q6,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_3268_minf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z2: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X3),Z2)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P3,X4)
                & aa_num_o(Q6,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_3269_minf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z2: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Z2)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P3,X4)
                & aa_nat_o(Q6,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_3270_minf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z2: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P3,X4)
                & aa_int_o(Q6,X4) ) ) ) ) ) ).

% minf(1)
tff(fact_3271_pinf_I7_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),X4) ) ).

% pinf(7)
tff(fact_3272_pinf_I7_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),X4) ) ).

% pinf(7)
tff(fact_3273_pinf_I7_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Ta),X4) ) ).

% pinf(7)
tff(fact_3274_pinf_I7_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ta),X4) ) ).

% pinf(7)
tff(fact_3275_pinf_I7_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4) ) ).

% pinf(7)
tff(fact_3276_pinf_I5_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Ta) ) ).

% pinf(5)
tff(fact_3277_pinf_I5_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Ta) ) ).

% pinf(5)
tff(fact_3278_pinf_I5_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Ta) ) ).

% pinf(5)
tff(fact_3279_pinf_I5_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Ta) ) ).

% pinf(5)
tff(fact_3280_pinf_I5_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta) ) ).

% pinf(5)
tff(fact_3281_pinf_I4_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_3282_pinf_I4_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_3283_pinf_I4_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_3284_pinf_I4_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_3285_pinf_I4_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(4)
tff(fact_3286_pinf_I3_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_3287_pinf_I3_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_3288_pinf_I3_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_3289_pinf_I3_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_3290_pinf_I3_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ( X4 != Ta ) ) ).

% pinf(3)
tff(fact_3291_pinf_I2_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z2: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
           => ( ( aa_real_o(P,X4)
                | aa_real_o(Q,X4) )
            <=> ( aa_real_o(P3,X4)
                | aa_real_o(Q6,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_3292_pinf_I2_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z2: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
           => ( ( aa_rat_o(P,X4)
                | aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P3,X4)
                | aa_rat_o(Q6,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_3293_pinf_I2_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z2: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
           => ( ( aa_num_o(P,X4)
                | aa_num_o(Q,X4) )
            <=> ( aa_num_o(P3,X4)
                | aa_num_o(Q6,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_3294_pinf_I2_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z2: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
           => ( ( aa_nat_o(P,X4)
                | aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P3,X4)
                | aa_nat_o(Q6,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_3295_pinf_I2_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z2: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
            <=> ( aa_int_o(P3,X4)
                | aa_int_o(Q6,X4) ) ) ) ) ) ).

% pinf(2)
tff(fact_3296_pinf_I1_J,axiom,
    ! [P: fun_real_o,P3: fun_real_o,Q: fun_real_o,Q6: fun_real_o] :
      ( ? [Z2: real] :
        ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
         => ( aa_real_o(P,X3)
          <=> aa_real_o(P3,X3) ) )
     => ( ? [Z2: real] :
          ! [X3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z2),X3)
           => ( aa_real_o(Q,X3)
            <=> aa_real_o(Q6,X3) ) )
       => ? [Z: real] :
          ! [X4: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
           => ( ( aa_real_o(P,X4)
                & aa_real_o(Q,X4) )
            <=> ( aa_real_o(P3,X4)
                & aa_real_o(Q6,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_3297_pinf_I1_J,axiom,
    ! [P: fun_rat_o,P3: fun_rat_o,Q: fun_rat_o,Q6: fun_rat_o] :
      ( ? [Z2: rat] :
        ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
         => ( aa_rat_o(P,X3)
          <=> aa_rat_o(P3,X3) ) )
     => ( ? [Z2: rat] :
          ! [X3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z2),X3)
           => ( aa_rat_o(Q,X3)
            <=> aa_rat_o(Q6,X3) ) )
       => ? [Z: rat] :
          ! [X4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
           => ( ( aa_rat_o(P,X4)
                & aa_rat_o(Q,X4) )
            <=> ( aa_rat_o(P3,X4)
                & aa_rat_o(Q6,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_3298_pinf_I1_J,axiom,
    ! [P: fun_num_o,P3: fun_num_o,Q: fun_num_o,Q6: fun_num_o] :
      ( ? [Z2: num] :
        ! [X3: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
         => ( aa_num_o(P,X3)
          <=> aa_num_o(P3,X3) ) )
     => ( ? [Z2: num] :
          ! [X3: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z2),X3)
           => ( aa_num_o(Q,X3)
            <=> aa_num_o(Q6,X3) ) )
       => ? [Z: num] :
          ! [X4: num] :
            ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
           => ( ( aa_num_o(P,X4)
                & aa_num_o(Q,X4) )
            <=> ( aa_num_o(P3,X4)
                & aa_num_o(Q6,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_3299_pinf_I1_J,axiom,
    ! [P: fun_nat_o,P3: fun_nat_o,Q: fun_nat_o,Q6: fun_nat_o] :
      ( ? [Z2: nat] :
        ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
         => ( aa_nat_o(P,X3)
          <=> aa_nat_o(P3,X3) ) )
     => ( ? [Z2: nat] :
          ! [X3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z2),X3)
           => ( aa_nat_o(Q,X3)
            <=> aa_nat_o(Q6,X3) ) )
       => ? [Z: nat] :
          ! [X4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
           => ( ( aa_nat_o(P,X4)
                & aa_nat_o(Q,X4) )
            <=> ( aa_nat_o(P3,X4)
                & aa_nat_o(Q6,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_3300_pinf_I1_J,axiom,
    ! [P: fun_int_o,P3: fun_int_o,Q: fun_int_o,Q6: fun_int_o] :
      ( ? [Z2: int] :
        ! [X3: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
         => ( aa_int_o(P,X3)
          <=> aa_int_o(P3,X3) ) )
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( aa_int_o(Q,X3)
            <=> aa_int_o(Q6,X3) ) )
       => ? [Z: int] :
          ! [X4: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
            <=> ( aa_int_o(P3,X4)
                & aa_int_o(Q6,X4) ) ) ) ) ) ).

% pinf(1)
tff(fact_3301_vebt__buildup_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero_nat )
     => ( ( Xb != aa_nat_nat(suc,zero_zero_nat) )
       => ~ ! [Va: nat] : Xb != aa_nat_nat(suc,aa_nat_nat(suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_3302_Euclid__induct,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: 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),B2) ) ) ) ).

% Euclid_induct
tff(fact_3303_minf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X4) ) ).

% minf(8)
tff(fact_3304_minf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X4) ) ).

% minf(8)
tff(fact_3305_minf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X4) ) ).

% minf(8)
tff(fact_3306_minf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X4) ) ).

% minf(8)
tff(fact_3307_minf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4) ) ).

% minf(8)
tff(fact_3308_minf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Ta) ) ).

% minf(6)
tff(fact_3309_minf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Ta) ) ).

% minf(6)
tff(fact_3310_minf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,X4),Z)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Ta) ) ).

% minf(6)
tff(fact_3311_minf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ta) ) ).

% minf(6)
tff(fact_3312_minf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta) ) ).

% minf(6)
tff(fact_3313_pinf_I8_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),X4) ) ).

% pinf(8)
tff(fact_3314_pinf_I8_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),X4) ) ).

% pinf(8)
tff(fact_3315_pinf_I8_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Ta),X4) ) ).

% pinf(8)
tff(fact_3316_pinf_I8_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Ta),X4) ) ).

% pinf(8)
tff(fact_3317_pinf_I8_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4) ) ).

% pinf(8)
tff(fact_3318_pinf_I6_J,axiom,
    ! [Ta: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),Ta) ) ).

% pinf(6)
tff(fact_3319_pinf_I6_J,axiom,
    ! [Ta: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,X4),Ta) ) ).

% pinf(6)
tff(fact_3320_pinf_I6_J,axiom,
    ! [Ta: num] :
    ? [Z: num] :
    ! [X4: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Z),X4)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X4),Ta) ) ).

% pinf(6)
tff(fact_3321_pinf_I6_J,axiom,
    ! [Ta: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ta) ) ).

% pinf(6)
tff(fact_3322_pinf_I6_J,axiom,
    ! [Ta: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta) ) ).

% pinf(6)
tff(fact_3323_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))) )
       => ! [X4: real,K4: real] :
            ( ( aa_real_o(P,X4)
              | aa_real_o(Q,X4) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))
              | aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3324_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,K2),D4))) )
       => ! [X4: rat,K4: rat] :
            ( ( aa_rat_o(P,X4)
              | aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))
              | aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3325_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(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
       => ! [X4: int,K4: int] :
            ( ( aa_int_o(P,X4)
              | aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(2)
tff(fact_3326_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))) )
       => ! [X4: real,K4: real] :
            ( ( aa_real_o(P,X4)
              & aa_real_o(Q,X4) )
          <=> ( aa_real_o(P,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4)))
              & aa_real_o(Q,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X4),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3327_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(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,K2),D4))) )
       => ! [X4: rat,K4: rat] :
            ( ( aa_rat_o(P,X4)
              & aa_rat_o(Q,X4) )
          <=> ( aa_rat_o(P,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4)))
              & aa_rat_o(Q,aa_rat_rat(minus_minus_rat(X4),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3328_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(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
     => ( ! [X3: int,K2: int] :
            ( aa_int_o(Q,X3)
          <=> aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
       => ! [X4: int,K4: int] :
            ( ( aa_int_o(P,X4)
              & aa_int_o(Q,X4) )
          <=> ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4)))
              & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),aa_int_int(aa_int_fun_int_int(times_times_int,K4),D4))) ) ) ) ) ).

% inf_period(1)
tff(fact_3329_dvd__productE,axiom,
    ! [P2: nat,A: nat,B2: 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),B2))
     => ~ ! [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),B2) ) ) ) ).

% dvd_productE
tff(fact_3330_dvd__productE,axiom,
    ! [P2: int,A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,P2),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2))
     => ~ ! [X3: int,Y3: int] :
            ( ( P2 = aa_int_int(aa_int_fun_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),B2) ) ) ) ).

% dvd_productE
tff(fact_3331_division__decomp,axiom,
    ! [A: nat,B2: 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,B2),C))
     => ? [B5: nat,C4: nat] :
          ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B5),C4) )
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B5),B2)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C4),C) ) ) ).

% division_decomp
tff(fact_3332_division__decomp,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C))
     => ? [B5: int,C4: int] :
          ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B5),C4) )
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B5),B2)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C4),C) ) ) ).

% division_decomp
tff(fact_3333_dvd__pos__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb) ) ) ).

% dvd_pos_nat
tff(fact_3334_gcd__nat__induct,axiom,
    ! [P: fun_nat_fun_nat_o,Mb: nat,Nb: nat] :
      ( ! [M3: nat] : aa_nat_o(aa_nat_fun_nat_o(P,M3),zero_zero_nat)
     => ( ! [M3: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( aa_nat_o(aa_nat_fun_nat_o(P,N2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M3),N2))
             => aa_nat_o(aa_nat_fun_nat_o(P,M3),N2) ) )
       => aa_nat_o(aa_nat_fun_nat_o(P,Mb),Nb) ) ) ).

% gcd_nat_induct
tff(fact_3335_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),X4)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)) ) ) ).

% pinf(9)
tff(fact_3336_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)) ) ) ).

% pinf(9)
tff(fact_3337_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)) ) ) ).

% pinf(9)
tff(fact_3338_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)) ) ) ).

% pinf(9)
tff(fact_3339_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)) ) ) ).

% pinf(9)
tff(fact_3340_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Z),X4)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)) ) ) ).

% pinf(10)
tff(fact_3341_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),X4)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)) ) ) ).

% pinf(10)
tff(fact_3342_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),X4)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)) ) ) ).

% pinf(10)
tff(fact_3343_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X4)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)) ) ) ).

% pinf(10)
tff(fact_3344_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X4)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)) ) ) ).

% pinf(10)
tff(fact_3345_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),Z)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)) ) ) ).

% minf(9)
tff(fact_3346_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))
      <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)) ) ) ).

% minf(9)
tff(fact_3347_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))
      <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)) ) ) ).

% minf(9)
tff(fact_3348_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)) ) ) ).

% minf(9)
tff(fact_3349_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)) ) ) ).

% minf(9)
tff(fact_3350_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z: code_integer] :
    ! [X4: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),Z)
     => ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S))
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,D),plus_p5714425477246183910nteger(X4,S)) ) ) ).

% minf(10)
tff(fact_3351_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z: real] :
    ! [X4: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,X4),Z)
     => ( ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S))
      <=> ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,D),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,X4),S)) ) ) ).

% minf(10)
tff(fact_3352_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z: rat] :
    ! [X4: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),Z)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S))
      <=> ~ aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,D),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),S)) ) ) ).

% minf(10)
tff(fact_3353_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z: nat] :
    ! [X4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Z)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S))
      <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),S)) ) ) ).

% minf(10)
tff(fact_3354_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z: int] :
    ! [X4: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Z)
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S))
      <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),S)) ) ) ).

% minf(10)
tff(fact_3355_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B2: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D),B2)
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Y)),D) )
            | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),Xb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y)),D) ) )
         => ? [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),B2))
              & ( ( 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),B2)),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),B2)),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_3356_bezout__add__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
      & ( ( 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,B2),Y3)),D2) )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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)),D2) ) ) ) ).

% bezout_add_nat
tff(fact_3357_bezout__add__strong__nat,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != zero_zero_nat )
     => ? [D2: nat,X3: nat,Y3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
          & ( 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,B2),Y3)),D2) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3358_bezout1__nat,axiom,
    ! [A: nat,B2: nat] :
    ? [D2: nat,X3: nat,Y3: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),A)
      & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),B2)
      & ( ( 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,B2),Y3)) = D2 )
        | ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),X3)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),Y3)) = D2 ) ) ) ).

% bezout1_nat
tff(fact_3359_Bolzano,axiom,
    ! [A: real,B2: real,P: fun_real_fun_real_o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [A4: real,B4: real,C3: real] :
            ( aa_real_o(aa_real_fun_real_o(P,A4),B4)
           => ( aa_real_o(aa_real_fun_real_o(P,B4),C3)
             => ( 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),C3)
                 => aa_real_o(aa_real_fun_real_o(P,A4),C3) ) ) ) )
       => ( ! [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),B2)
               => ? [D5: real] :
                    ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D5)
                    & ! [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)),D5) )
                       => aa_real_o(aa_real_fun_real_o(P,A4),B4) ) ) ) )
         => aa_real_o(aa_real_fun_real_o(P,A),B2) ) ) ) ).

% Bolzano
tff(fact_3360_mult__le__cancel__iff1,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Zb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3361_mult__le__cancel__iff1,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Zb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3362_mult__le__cancel__iff1,axiom,
    ! [Zb: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Zb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Zb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_3363_mult__le__cancel__iff2,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Zb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3364_mult__le__cancel__iff2,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Zb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),Y))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3365_mult__le__cancel__iff2,axiom,
    ! [Zb: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Zb),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,Zb),Y))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_3366_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_3367_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_3368_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ).

% signed_take_bit_rec
tff(fact_3369_verit__minus__simplify_I4_J,axiom,
    ! [B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(uminus_uminus_real,B2)) = B2 ).

% verit_minus_simplify(4)
tff(fact_3370_verit__minus__simplify_I4_J,axiom,
    ! [B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,B2)) = B2 ).

% verit_minus_simplify(4)
tff(fact_3371_verit__minus__simplify_I4_J,axiom,
    ! [B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,B2)) = B2 ).

% verit_minus_simplify(4)
tff(fact_3372_verit__minus__simplify_I4_J,axiom,
    ! [B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = B2 ).

% verit_minus_simplify(4)
tff(fact_3373_verit__minus__simplify_I4_J,axiom,
    ! [B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(uminus_uminus_rat,B2)) = B2 ).

% verit_minus_simplify(4)
tff(fact_3374_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_3375_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_3376_add_Oinverse__inverse,axiom,
    ! [A: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = A ).

% add.inverse_inverse
tff(fact_3377_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_3378_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_3379_neg__equal__iff__equal,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3380_neg__equal__iff__equal,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3381_neg__equal__iff__equal,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3382_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3383_neg__equal__iff__equal,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( A = B2 ) ) ).

% neg_equal_iff_equal
tff(fact_3384_neg__le__iff__le,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% neg_le_iff_le
tff(fact_3385_neg__le__iff__le,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2) ) ).

% neg_le_iff_le
tff(fact_3386_neg__le__iff__le,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% neg_le_iff_le
tff(fact_3387_neg__le__iff__le,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% neg_le_iff_le
tff(fact_3388_add_Oinverse__neutral,axiom,
    aa_real_real(uminus_uminus_real,zero_zero_real) = zero_zero_real ).

% add.inverse_neutral
tff(fact_3389_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_3390_add_Oinverse__neutral,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,zero_zero_complex) = zero_zero_complex ).

% add.inverse_neutral
tff(fact_3391_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_3392_add_Oinverse__neutral,axiom,
    aa_rat_rat(uminus_uminus_rat,zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_3393_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_3394_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_3395_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_3396_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_3397_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_3398_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_3399_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_3400_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_3401_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_3402_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_3403_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A = aa_real_real(uminus_uminus_real,A) )
    <=> ( A = zero_zero_real ) ) ).

% equal_neg_zero
tff(fact_3404_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_3405_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_3406_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_3407_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = A )
    <=> ( A = zero_zero_real ) ) ).

% neg_equal_zero
tff(fact_3408_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_3409_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_3410_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_3411_neg__less__iff__less,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% neg_less_iff_less
tff(fact_3412_neg__less__iff__less,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% neg_less_iff_less
tff(fact_3413_neg__less__iff__less,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2) ) ).

% neg_less_iff_less
tff(fact_3414_neg__less__iff__less,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% neg_less_iff_less
tff(fact_3415_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3416_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3417_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3418_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3419_neg__numeral__eq__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) )
    <=> ( Mb = Nb ) ) ).

% neg_numeral_eq_iff
tff(fact_3420_mult__minus__right,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ).

% mult_minus_right
tff(fact_3421_mult__minus__right,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ).

% mult_minus_right
tff(fact_3422_mult__minus__right,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B2)) ).

% mult_minus_right
tff(fact_3423_mult__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) ).

% mult_minus_right
tff(fact_3424_mult__minus__right,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ).

% mult_minus_right
tff(fact_3425_minus__mult__minus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) ).

% minus_mult_minus
tff(fact_3426_minus__mult__minus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B2) ).

% minus_mult_minus
tff(fact_3427_minus__mult__minus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(times_times_complex(A),B2) ).

% minus_mult_minus
tff(fact_3428_minus__mult__minus,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2) ).

% minus_mult_minus
tff(fact_3429_minus__mult__minus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) ).

% minus_mult_minus
tff(fact_3430_mult__minus__left,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ).

% mult_minus_left
tff(fact_3431_mult__minus__left,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ).

% mult_minus_left
tff(fact_3432_mult__minus__left,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(A),B2)) ).

% mult_minus_left
tff(fact_3433_mult__minus__left,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) ).

% mult_minus_left
tff(fact_3434_mult__minus__left,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) ).

% mult_minus_left
tff(fact_3435_minus__add__distrib,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) ).

% minus_add_distrib
tff(fact_3436_minus__add__distrib,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) ).

% minus_add_distrib
tff(fact_3437_minus__add__distrib,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_add_distrib
tff(fact_3438_minus__add__distrib,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% minus_add_distrib
tff(fact_3439_minus__add__distrib,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_add_distrib
tff(fact_3440_minus__add__cancel,axiom,
    ! [A: real,B2: 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),B2)) = B2 ).

% minus_add_cancel
tff(fact_3441_minus__add__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = B2 ).

% minus_add_cancel
tff(fact_3442_minus__add__cancel,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(plus_plus_complex(A),B2)) = B2 ).

% minus_add_cancel
tff(fact_3443_minus__add__cancel,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),plus_p5714425477246183910nteger(A,B2)) = B2 ).

% minus_add_cancel
tff(fact_3444_minus__add__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = B2 ).

% minus_add_cancel
tff(fact_3445_add__minus__cancel,axiom,
    ! [A: real,B2: 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)),B2)) = B2 ).

% add_minus_cancel
tff(fact_3446_add__minus__cancel,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_3447_add__minus__cancel,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(A),aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_3448_add__minus__cancel,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(A,plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2)) = B2 ).

% add_minus_cancel
tff(fact_3449_add__minus__cancel,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)) = B2 ).

% add_minus_cancel
tff(fact_3450_minus__diff__eq,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A) ).

% minus_diff_eq
tff(fact_3451_minus__diff__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% minus_diff_eq
tff(fact_3452_minus__diff__eq,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B2)) = aa_complex_complex(minus_minus_complex(B2),A) ).

% minus_diff_eq
tff(fact_3453_minus__diff__eq,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B2)) = minus_8373710615458151222nteger(B2,A) ).

% minus_diff_eq
tff(fact_3454_minus__diff__eq,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B2)) = aa_rat_rat(minus_minus_rat(B2),A) ).

% minus_diff_eq
tff(fact_3455_div__minus__minus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ).

% div_minus_minus
tff(fact_3456_div__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = divide6298287555418463151nteger(A,B2) ).

% div_minus_minus
tff(fact_3457_dvd__minus__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),aa_real_real(uminus_uminus_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3458_dvd__minus__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),aa_int_int(uminus_uminus_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3459_dvd__minus__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,Y))
    <=> aa_complex_o(dvd_dvd_complex(Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3460_dvd__minus__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3461_dvd__minus__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),aa_rat_rat(uminus_uminus_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% dvd_minus_iff
tff(fact_3462_minus__dvd__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(uminus_uminus_real,Xb)),Y)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3463_minus__dvd__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(uminus_uminus_int,Xb)),Y)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3464_minus__dvd__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( aa_complex_o(dvd_dvd_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)
    <=> aa_complex_o(dvd_dvd_complex(Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3465_minus__dvd__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)),Y)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3466_minus__dvd__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(uminus_uminus_rat,Xb)),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Xb),Y) ) ).

% minus_dvd_iff
tff(fact_3467_mod__minus__minus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)) ).

% mod_minus_minus
tff(fact_3468_mod__minus__minus,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B2)) ).

% mod_minus_minus
tff(fact_3469_real__add__minus__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,A)) = zero_zero_real )
    <=> ( Xb = A ) ) ).

% real_add_minus_iff
tff(fact_3470_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_3471_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_3472_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_3473_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_3474_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_3475_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_3476_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_3477_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_3478_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_3479_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_3480_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_3481_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_3482_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_3483_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_3484_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_3485_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_3486_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_3487_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_3488_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_3489_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_3490_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_3491_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_3492_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_3493_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_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_3504_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_3505_ab__left__minus,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_3506_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_3507_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_3508_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_3509_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_3510_add_Oright__inverse,axiom,
    ! [A: code_integer] : plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_3511_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_3512_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_3513_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_3514_diff__0,axiom,
    ! [A: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),A) = aa_complex_complex(uminus1482373934393186551omplex,A) ).

% diff_0
tff(fact_3515_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_3516_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_3517_verit__minus__simplify_I3_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,zero_zero_real),B2) = aa_real_real(uminus_uminus_real,B2) ).

% verit_minus_simplify(3)
tff(fact_3518_verit__minus__simplify_I3_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),B2) = aa_int_int(uminus_uminus_int,B2) ).

% verit_minus_simplify(3)
tff(fact_3519_verit__minus__simplify_I3_J,axiom,
    ! [B2: complex] : aa_complex_complex(minus_minus_complex(zero_zero_complex),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% verit_minus_simplify(3)
tff(fact_3520_verit__minus__simplify_I3_J,axiom,
    ! [B2: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% verit_minus_simplify(3)
tff(fact_3521_verit__minus__simplify_I3_J,axiom,
    ! [B2: rat] : aa_rat_rat(minus_minus_rat(zero_zero_rat),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% verit_minus_simplify(3)
tff(fact_3522_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),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,Mb)),aa_num_real(numeral_numeral_real,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3523_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3524_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Mb)),numera6690914467698888265omplex(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3525_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3526_add__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_num_rat(numeral_numeral_rat,Nb))) ).

% add_neg_numeral_simps(3)
tff(fact_3527_mult__minus1,axiom,
    ! [Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,one_one_real)),Zb) = aa_real_real(uminus_uminus_real,Zb) ).

% mult_minus1
tff(fact_3528_mult__minus1,axiom,
    ! [Zb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,one_one_int)),Zb) = aa_int_int(uminus_uminus_int,Zb) ).

% mult_minus1
tff(fact_3529_mult__minus1,axiom,
    ! [Zb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Zb) = aa_complex_complex(uminus1482373934393186551omplex,Zb) ).

% mult_minus1
tff(fact_3530_mult__minus1,axiom,
    ! [Zb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Zb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Zb) ).

% mult_minus1
tff(fact_3531_mult__minus1,axiom,
    ! [Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Zb) = aa_rat_rat(uminus_uminus_rat,Zb) ).

% mult_minus1
tff(fact_3532_mult__minus1__right,axiom,
    ! [Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Zb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Zb) ).

% mult_minus1_right
tff(fact_3533_mult__minus1__right,axiom,
    ! [Zb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Zb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,Zb) ).

% mult_minus1_right
tff(fact_3534_mult__minus1__right,axiom,
    ! [Zb: complex] : aa_complex_complex(times_times_complex(Zb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Zb) ).

% mult_minus1_right
tff(fact_3535_mult__minus1__right,axiom,
    ! [Zb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Zb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Zb) ).

% mult_minus1_right
tff(fact_3536_mult__minus1__right,axiom,
    ! [Zb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Zb) ).

% mult_minus1_right
tff(fact_3537_uminus__add__conv__diff,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A) ).

% uminus_add_conv_diff
tff(fact_3538_uminus__add__conv__diff,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A) ).

% uminus_add_conv_diff
tff(fact_3539_uminus__add__conv__diff,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(minus_minus_complex(B2),A) ).

% uminus_add_conv_diff
tff(fact_3540_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B2: code_integer] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) = minus_8373710615458151222nteger(B2,A) ).

% uminus_add_conv_diff
tff(fact_3541_uminus__add__conv__diff,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) = aa_rat_rat(minus_minus_rat(B2),A) ).

% uminus_add_conv_diff
tff(fact_3542_diff__minus__eq__add,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) ).

% diff_minus_eq_add
tff(fact_3543_diff__minus__eq__add,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) ).

% diff_minus_eq_add
tff(fact_3544_diff__minus__eq__add,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(plus_plus_complex(A),B2) ).

% diff_minus_eq_add
tff(fact_3545_diff__minus__eq__add,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = plus_p5714425477246183910nteger(A,B2) ).

% diff_minus_eq_add
tff(fact_3546_diff__minus__eq__add,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) ).

% diff_minus_eq_add
tff(fact_3547_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_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_3548_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_3549_divide__minus1,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,Xb) ).

% divide_minus1
tff(fact_3550_divide__minus1,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% divide_minus1
tff(fact_3551_divide__minus1,axiom,
    ! [Xb: rat] : aa_rat_rat(divide_divide_rat(Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,Xb) ).

% divide_minus1
tff(fact_3552_minus__mod__self1,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B2) ).

% minus_mod_self1
tff(fact_3553_minus__mod__self1,axiom,
    ! [B2: code_integer,A: code_integer] : modulo364778990260209775nteger(minus_8373710615458151222nteger(B2,A),B2) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) ).

% minus_mod_self1
tff(fact_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_add__neg__numeral__special_I8_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = zero_zero_int ).

% add_neg_numeral_special(8)
tff(fact_3564_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_3565_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_3566_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_3567_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_3568_add__neg__numeral__special_I7_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% add_neg_numeral_special(7)
tff(fact_3569_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_3570_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_3571_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_3572_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_3573_diff__numeral__special_I12_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% diff_numeral_special(12)
tff(fact_3574_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_3575_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_3576_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_3577_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_3578_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_3579_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_3580_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_3581_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_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)) = one_one_int ).

% minus_one_mult_self
tff(fact_3589_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_3590_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_3591_minus__one__mult__self,axiom,
    ! [Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),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_3592_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_3593_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3594_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_3595_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3596_left__minus__one__mult__self,axiom,
    ! [Nb: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),A)) = A ).

% left_minus_one_mult_self
tff(fact_3597_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_3598_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_3599_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(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3600_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3601_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(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3602_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(plus_plus_num(V),W))),Y) ).

% semiring_norm(168)
tff(fact_3603_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(V),W)))),Y) ).

% semiring_norm(168)
tff(fact_3604_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3605_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3606_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3607_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3608_diff__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb))) ).

% diff_numeral_simps(3)
tff(fact_3609_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3610_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3611_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3612_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3613_diff__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),Nb)) ).

% diff_numeral_simps(2)
tff(fact_3614_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(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_3615_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_3616_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(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_3617_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(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_3618_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W))),Y) ).

% semiring_norm(172)
tff(fact_3619_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(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3620_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,V)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3621_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(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3622_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(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3623_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,V)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(171)
tff(fact_3624_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(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3625_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3626_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(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3627_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(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3628_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(V),W)))),Y) ).

% semiring_norm(170)
tff(fact_3629_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Mb)),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(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3630_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3631_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3632_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3633_mult__neg__numeral__simps_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,Mb)),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(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(3)
tff(fact_3634_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3635_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3636_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),numera6690914467698888265omplex(Nb)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3637_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3638_mult__neg__numeral__simps_I2_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb))) ).

% mult_neg_numeral_simps(2)
tff(fact_3639_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) = aa_num_real(numeral_numeral_real,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3640_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3641_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb))) = numera6690914467698888265omplex(aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3642_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3643_mult__neg__numeral__simps_I1_J,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) = aa_num_rat(numeral_numeral_rat,aa_num_num(times_times_num(Mb),Nb)) ).

% mult_neg_numeral_simps(1)
tff(fact_3644_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),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),Mb) ) ).

% neg_numeral_le_iff
tff(fact_3645_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_3646_neg__numeral__le__iff,axiom,
    ! [Mb: 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,Mb))),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),Mb) ) ).

% neg_numeral_le_iff
tff(fact_3647_neg__numeral__le__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Nb),Mb) ) ).

% neg_numeral_le_iff
tff(fact_3648_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),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),Mb) ) ).

% neg_numeral_less_iff
tff(fact_3649_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_3650_neg__numeral__less__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Nb),Mb) ) ).

% neg_numeral_less_iff
tff(fact_3651_neg__numeral__less__iff,axiom,
    ! [Mb: 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,Mb))),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),Mb) ) ).

% neg_numeral_less_iff
tff(fact_3652_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3653_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3654_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: 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,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3655_not__neg__one__le__neg__numeral__iff,axiom,
    ! [Mb: num] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)))
    <=> ( Mb != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3656_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: 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,B2),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)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3657_divide__le__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),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(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B2) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3658_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: 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,B2),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,B2),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_3659_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_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_3660_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: real,W: num,A: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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,B2 = 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_3661_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: complex,W: num,A: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = A )
    <=> $ite(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) != zero_zero_complex,B2 = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),A = zero_zero_complex) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3662_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( ( aa_rat_rat(divide_divide_rat(B2),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,B2 = aa_rat_rat(aa_rat_fun_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_3663_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( A = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),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))) = B2,A = zero_zero_real) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3664_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B2: complex,W: num] :
      ( ( A = aa_complex_complex(divide1717551699836669952omplex(B2),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))) = B2,A = zero_zero_complex) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3665_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( ( A = aa_rat_rat(divide_divide_rat(B2),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(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = B2,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3666_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3667_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3668_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3669_neg__numeral__less__neg__one__iff,axiom,
    ! [Mb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat))
    <=> ( Mb != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3670_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: 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,B2),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)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3671_divide__less__eq__numeral1_I2_J,axiom,
    ! [B2: rat,W: num,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),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(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)))),B2) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3672_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B2: real,W: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_3673_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B2: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(divide_divide_rat(B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(aa_rat_fun_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_3674_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_3675_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_3676_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_3677_power2__minus,axiom,
    ! [A: code_integer] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(A,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_minus
tff(fact_3678_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_3679_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_3680_add__neg__numeral__special_I9_J,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_3681_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_3682_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_3683_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ).

% add_neg_numeral_special(9)
tff(fact_3684_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_3685_diff__numeral__special_I11_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% diff_numeral_special(11)
tff(fact_3686_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_3687_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_3688_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_3689_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_3690_diff__numeral__special_I10_J,axiom,
    aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% diff_numeral_special(10)
tff(fact_3691_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_3692_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_3693_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_3694_minus__1__div__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_3695_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_3696_minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_3697_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_3698_bits__minus__1__mod__2__eq,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_3699_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_3700_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_3701_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_3702_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_3703_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,Nb: nat] : 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)) = 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_3704_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_3705_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_3706_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_3707_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_3708_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(A,Nb)) ) ) ).

% power_minus_odd
tff(fact_3709_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_3710_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_3711_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_3712_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_3713_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb) = power_8256067586552552935nteger(A,Nb) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_3714_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_3715_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(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3716_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3717_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(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3718_diff__numeral__special_I3_J,axiom,
    ! [Nb: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3719_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(plus_plus_num(one),Nb)) ).

% diff_numeral_special(3)
tff(fact_3720_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_3721_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_3722_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),one_one_complex) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_3723_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_3724_diff__numeral__special_I4_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(plus_plus_num(Mb),one))) ).

% diff_numeral_special(4)
tff(fact_3725_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(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_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_3734_power__minus1__even,axiom,
    ! [Nb: nat] : 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_3735_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_3736_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_3737_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_3738_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_3739_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_3740_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_3741_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_3742_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_3743_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_3744_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_3745_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_3746_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_3747_signed__take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_ri631733984087533419it_int(zero_zero_nat),A) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% signed_take_bit_0
tff(fact_3748_verit__negate__coefficient_I3_J,axiom,
    ! [A: real,B2: real] :
      ( ( A = B2 )
     => ( aa_real_real(uminus_uminus_real,A) = aa_real_real(uminus_uminus_real,B2) ) ) ).

% verit_negate_coefficient(3)
tff(fact_3749_verit__negate__coefficient_I3_J,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
     => ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B2) ) ) ).

% verit_negate_coefficient(3)
tff(fact_3750_verit__negate__coefficient_I3_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = B2 )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ) ) ).

% verit_negate_coefficient(3)
tff(fact_3751_verit__negate__coefficient_I3_J,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
     => ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(uminus_uminus_rat,B2) ) ) ).

% verit_negate_coefficient(3)
tff(fact_3752_equation__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% equation_minus_iff
tff(fact_3753_equation__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B2) )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_3754_equation__minus__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% equation_minus_iff
tff(fact_3755_equation__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_3756_equation__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% equation_minus_iff
tff(fact_3757_minus__equation__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(uminus_uminus_real,B2) = A ) ) ).

% minus_equation_iff
tff(fact_3758_minus__equation__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B2 )
    <=> ( aa_int_int(uminus_uminus_int,B2) = A ) ) ).

% minus_equation_iff
tff(fact_3759_minus__equation__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 )
    <=> ( aa_complex_complex(uminus1482373934393186551omplex,B2) = A ) ) ).

% minus_equation_iff
tff(fact_3760_minus__equation__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) = A ) ) ).

% minus_equation_iff
tff(fact_3761_minus__equation__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B2 )
    <=> ( aa_rat_rat(uminus_uminus_rat,B2) = A ) ) ).

% minus_equation_iff
tff(fact_3762_le__imp__neg__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ) ).

% le_imp_neg_le
tff(fact_3763_le__imp__neg__le,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_3764_le__imp__neg__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_imp_neg_le
tff(fact_3765_le__imp__neg__le,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_3766_minus__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),A) ) ).

% minus_le_iff
tff(fact_3767_minus__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),A) ) ).

% minus_le_iff
tff(fact_3768_minus__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),A) ) ).

% minus_le_iff
tff(fact_3769_minus__le__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B2)),A) ) ).

% minus_le_iff
tff(fact_3770_le__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),aa_real_real(uminus_uminus_real,A)) ) ).

% le_minus_iff
tff(fact_3771_le__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_3772_le__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(uminus_uminus_rat,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% le_minus_iff
tff(fact_3773_le__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_3774_minus__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),A) ) ).

% minus_less_iff
tff(fact_3775_minus__less__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),A) ) ).

% minus_less_iff
tff(fact_3776_minus__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),A) ) ).

% minus_less_iff
tff(fact_3777_minus__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),A) ) ).

% minus_less_iff
tff(fact_3778_less__minus__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(uminus_uminus_real,B2))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),aa_real_real(uminus_uminus_real,A)) ) ).

% less_minus_iff
tff(fact_3779_less__minus__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_3780_less__minus__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_3781_less__minus__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(uminus_uminus_rat,B2))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% less_minus_iff
tff(fact_3782_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3783_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3784_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3785_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ) ).

% verit_negate_coefficient(2)
tff(fact_3786_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb)) != aa_num_real(numeral_numeral_real,Nb) ).

% neg_numeral_neq_numeral
tff(fact_3787_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)) != aa_num_int(numeral_numeral_int,Nb) ).

% neg_numeral_neq_numeral
tff(fact_3788_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb)) != numera6690914467698888265omplex(Nb) ).

% neg_numeral_neq_numeral
tff(fact_3789_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)) != numera6620942414471956472nteger(Nb) ).

% neg_numeral_neq_numeral
tff(fact_3790_neg__numeral__neq__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb)) != aa_num_rat(numeral_numeral_rat,Nb) ).

% neg_numeral_neq_numeral
tff(fact_3791_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_real(numeral_numeral_real,Mb) != aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_3792_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,Mb) != aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_3793_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6690914467698888265omplex(Mb) != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_3794_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : numera6620942414471956472nteger(Mb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% numeral_neq_neg_numeral
tff(fact_3795_numeral__neq__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_num_rat(numeral_numeral_rat,Mb) != aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb)) ).

% numeral_neq_neg_numeral
tff(fact_3796_minus__mult__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% minus_mult_commute
tff(fact_3797_minus__mult__commute,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% minus_mult_commute
tff(fact_3798_minus__mult__commute,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_mult_commute
tff(fact_3799_minus__mult__commute,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% minus_mult_commute
tff(fact_3800_minus__mult__commute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_mult_commute
tff(fact_3801_square__eq__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% square_eq_iff
tff(fact_3802_square__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_int_int(uminus_uminus_int,B2) ) ) ) ).

% square_eq_iff
tff(fact_3803_square__eq__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),A) = aa_complex_complex(times_times_complex(B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% square_eq_iff
tff(fact_3804_square__eq__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ) ) ) ).

% square_eq_iff
tff(fact_3805_square__eq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),B2) )
    <=> ( ( A = B2 )
        | ( A = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ).

% square_eq_iff
tff(fact_3806_one__neq__neg__one,axiom,
    one_one_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% one_neq_neg_one
tff(fact_3807_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_3808_one__neq__neg__one,axiom,
    one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% one_neq_neg_one
tff(fact_3809_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_3810_one__neq__neg__one,axiom,
    one_one_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% one_neq_neg_one
tff(fact_3811_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% add.inverse_distrib_swap
tff(fact_3812_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_3813_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% add.inverse_distrib_swap
tff(fact_3814_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_3815_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ).

% add.inverse_distrib_swap
tff(fact_3816_group__cancel_Oneg1,axiom,
    ! [A3: real,K: real,A: real] :
      ( ( A3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),A) )
     => ( aa_real_real(uminus_uminus_real,A3) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,K)),aa_real_real(uminus_uminus_real,A)) ) ) ).

% group_cancel.neg1
tff(fact_3817_group__cancel_Oneg1,axiom,
    ! [A3: int,K: int,A: int] :
      ( ( A3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_3818_group__cancel_Oneg1,axiom,
    ! [A3: complex,K: complex,A: complex] :
      ( ( A3 = aa_complex_complex(plus_plus_complex(K),A) )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(uminus1482373934393186551omplex,A)) ) ) ).

% group_cancel.neg1
tff(fact_3819_group__cancel_Oneg1,axiom,
    ! [A3: code_integer,K: code_integer,A: code_integer] :
      ( ( A3 = plus_p5714425477246183910nteger(K,A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_3820_group__cancel_Oneg1,axiom,
    ! [A3: rat,K: rat,A: rat] :
      ( ( A3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(uminus_uminus_rat,A3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(uminus_uminus_rat,A)) ) ) ).

% group_cancel.neg1
tff(fact_3821_is__num__normalize_I8_J,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(uminus_uminus_real,A)) ).

% is_num_normalize(8)
tff(fact_3822_is__num__normalize_I8_J,axiom,
    ! [A: int,B2: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B2)),aa_int_int(uminus_uminus_int,A)) ).

% is_num_normalize(8)
tff(fact_3823_is__num__normalize_I8_J,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(plus_plus_complex(A),B2)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),aa_complex_complex(uminus1482373934393186551omplex,A)) ).

% is_num_normalize(8)
tff(fact_3824_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% is_num_normalize(8)
tff(fact_3825_is__num__normalize_I8_J,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(uminus_uminus_rat,A)) ).

% is_num_normalize(8)
tff(fact_3826_minus__diff__commute,axiom,
    ! [B2: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,B2)),A) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),B2) ).

% minus_diff_commute
tff(fact_3827_minus__diff__commute,axiom,
    ! [B2: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B2)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B2) ).

% minus_diff_commute
tff(fact_3828_minus__diff__commute,axiom,
    ! [B2: complex,A: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,B2)),A) = aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) ).

% minus_diff_commute
tff(fact_3829_minus__diff__commute,axiom,
    ! [B2: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) ).

% minus_diff_commute
tff(fact_3830_minus__diff__commute,axiom,
    ! [B2: rat,A: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,B2)),A) = aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),B2) ).

% minus_diff_commute
tff(fact_3831_minus__diff__minus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) ).

% minus_diff_minus
tff(fact_3832_minus__diff__minus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ).

% minus_diff_minus
tff(fact_3833_minus__diff__minus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(minus_minus_complex(A),B2)) ).

% minus_diff_minus
tff(fact_3834_minus__diff__minus,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B2)) ).

% minus_diff_minus
tff(fact_3835_minus__diff__minus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(minus_minus_rat(A),B2)) ).

% minus_diff_minus
tff(fact_3836_div__minus__right,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B2) ).

% div_minus_right
tff(fact_3837_div__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) ).

% div_minus_right
tff(fact_3838_minus__divide__right,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% minus_divide_right
tff(fact_3839_minus__divide__right,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% minus_divide_right
tff(fact_3840_minus__divide__right,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% minus_divide_right
tff(fact_3841_minus__divide__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ).

% minus_divide_divide
tff(fact_3842_minus__divide__divide,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ).

% minus_divide_divide
tff(fact_3843_minus__divide__divide,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(divide_divide_rat(A),B2) ).

% minus_divide_divide
tff(fact_3844_minus__divide__left,axiom,
    ! [A: real,B2: real] : aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B2) ).

% minus_divide_left
tff(fact_3845_minus__divide__left,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) ).

% minus_divide_left
tff(fact_3846_minus__divide__left,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B2) ).

% minus_divide_left
tff(fact_3847_mod__minus__eq,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B2) ).

% mod_minus_eq
tff(fact_3848_mod__minus__eq,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,B2)),B2) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) ).

% mod_minus_eq
tff(fact_3849_mod__minus__cong,axiom,
    ! [A: int,B2: int,A2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A2),B2) )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A2)),B2) ) ) ).

% mod_minus_cong
tff(fact_3850_mod__minus__cong,axiom,
    ! [A: code_integer,B2: code_integer,A2: code_integer] :
      ( ( modulo364778990260209775nteger(A,B2) = modulo364778990260209775nteger(A2,B2) )
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2),B2) ) ) ).

% mod_minus_cong
tff(fact_3851_mod__minus__right,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B2)) ).

% mod_minus_right
tff(fact_3852_mod__minus__right,axiom,
    ! [A: code_integer,B2: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2)) ).

% mod_minus_right
tff(fact_3853_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_3854_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_le_numeral
tff(fact_3855_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_le_numeral
tff(fact_3856_neg__numeral__le__numeral,axiom,
    ! [Mb: 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,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_le_numeral
tff(fact_3857_neg__numeral__le__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_le_numeral
tff(fact_3858_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_3859_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_3860_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_3861_not__numeral__le__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_le_neg_numeral
tff(fact_3862_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_3863_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_3864_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% zero_neq_neg_numeral
tff(fact_3865_zero__neq__neg__numeral,axiom,
    ! [Nb: num] : zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% zero_neq_neg_numeral
tff(fact_3866_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_3867_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_3868_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_3869_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_3870_not__numeral__less__neg__numeral,axiom,
    ! [Mb: num,Nb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Nb))) ).

% not_numeral_less_neg_numeral
tff(fact_3871_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_num_real(numeral_numeral_real,Nb)) ).

% neg_numeral_less_numeral
tff(fact_3872_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) ).

% neg_numeral_less_numeral
tff(fact_3873_neg__numeral__less__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),numera6620942414471956472nteger(Nb)) ).

% neg_numeral_less_numeral
tff(fact_3874_neg__numeral__less__numeral,axiom,
    ! [Mb: 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,Mb))),aa_num_rat(numeral_numeral_rat,Nb)) ).

% neg_numeral_less_numeral
tff(fact_3875_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_3876_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_3877_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_3878_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_3879_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_3880_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_3881_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_3882_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_3883_zero__neq__neg__one,axiom,
    zero_zero_real != aa_real_real(uminus_uminus_real,one_one_real) ).

% zero_neq_neg_one
tff(fact_3884_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_3885_zero__neq__neg__one,axiom,
    zero_zero_complex != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% zero_neq_neg_one
tff(fact_3886_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_3887_zero__neq__neg__one,axiom,
    zero_zero_rat != aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% zero_neq_neg_one
tff(fact_3888_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(uminus_uminus_real,A) = B2 )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3889_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B2 )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3890_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3891_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 )
    <=> ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3892_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,A) = B2 )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3893_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(uminus_uminus_real,B2) )
    <=> ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3894_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B2) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3895_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) )
    <=> ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3896_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) )
    <=> ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3897_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,B2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3898_add_Oinverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_3899_add_Oinverse__unique,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_3900_add_Oinverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_3901_add_Oinverse__unique,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_3902_add_Oinverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,A) = B2 ) ) ).

% add.inverse_unique
tff(fact_3903_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_3904_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_3905_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_3906_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_3907_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_3908_add__eq__0__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2) = zero_zero_real )
    <=> ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ).

% add_eq_0_iff
tff(fact_3909_add__eq__0__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = zero_zero_int )
    <=> ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_3910_add__eq__0__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(plus_plus_complex(A),B2) = zero_zero_complex )
    <=> ( B2 = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ).

% add_eq_0_iff
tff(fact_3911_add__eq__0__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( plus_p5714425477246183910nteger(A,B2) = zero_z3403309356797280102nteger )
    <=> ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_3912_add__eq__0__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = zero_zero_rat )
    <=> ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ).

% add_eq_0_iff
tff(fact_3913_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_3914_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_3915_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_3916_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_3917_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_3918_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_3919_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_3920_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_3921_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% numeral_times_minus_swap
tff(fact_3922_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,W)),aa_int_int(uminus_uminus_int,Xb)) = aa_int_int(aa_int_fun_int_int(times_times_int,Xb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,W))) ).

% numeral_times_minus_swap
tff(fact_3923_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: complex] : aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% numeral_times_minus_swap
tff(fact_3924_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_3925_numeral__times__minus__swap,axiom,
    ! [W: num,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,W)),aa_rat_rat(uminus_uminus_rat,Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% numeral_times_minus_swap
tff(fact_3926_nonzero__minus__divide__right,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3927_nonzero__minus__divide__right,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3928_nonzero__minus__divide__right,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% nonzero_minus_divide_right
tff(fact_3929_nonzero__minus__divide__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != 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,B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3930_nonzero__minus__divide__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(divide1717551699836669952omplex(A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3931_nonzero__minus__divide__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(divide_divide_rat(A),B2) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3932_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_3933_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_3934_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_complex != aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Nb)) ).

% one_neq_neg_numeral
tff(fact_3935_one__neq__neg__numeral,axiom,
    ! [Nb: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb)) ).

% one_neq_neg_numeral
tff(fact_3936_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_3937_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_3938_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_3939_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6690914467698888265omplex(Nb) != aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% numeral_neq_neg_one
tff(fact_3940_numeral__neq__neg__one,axiom,
    ! [Nb: num] : numera6620942414471956472nteger(Nb) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_3941_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_3942_square__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb) = one_one_real )
    <=> ( ( Xb = one_one_real )
        | ( Xb = aa_real_real(uminus_uminus_real,one_one_real) ) ) ) ).

% square_eq_1_iff
tff(fact_3943_square__eq__1__iff,axiom,
    ! [Xb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Xb) = one_one_int )
    <=> ( ( Xb = one_one_int )
        | ( Xb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_3944_square__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Xb) = one_one_complex )
    <=> ( ( Xb = one_one_complex )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ) ).

% square_eq_1_iff
tff(fact_3945_square__eq__1__iff,axiom,
    ! [Xb: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xb) = one_one_Code_integer )
    <=> ( ( Xb = one_one_Code_integer )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_3946_square__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Xb) = one_one_rat )
    <=> ( ( Xb = one_one_rat )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_3947_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3948_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3949_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3950_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3951_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3952_diff__conv__add__uminus,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_real_real(uminus_uminus_real,B2)) ).

% diff_conv_add_uminus
tff(fact_3953_diff__conv__add__uminus,axiom,
    ! [A: int,B2: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B2)) ).

% diff_conv_add_uminus
tff(fact_3954_diff__conv__add__uminus,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(minus_minus_complex(A),B2) = aa_complex_complex(plus_plus_complex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) ).

% diff_conv_add_uminus
tff(fact_3955_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B2: code_integer] : minus_8373710615458151222nteger(A,B2) = plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) ).

% diff_conv_add_uminus
tff(fact_3956_diff__conv__add__uminus,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(minus_minus_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(uminus_uminus_rat,B2)) ).

% diff_conv_add_uminus
tff(fact_3957_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B2: real,A: real] :
      ( ( B3 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,K),B2) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B3) = 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),B2)) ) ) ).

% group_cancel.sub2
tff(fact_3958_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B2: int,A: int] :
      ( ( B3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B3) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_3959_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B2: complex,A: complex] :
      ( ( B3 = aa_complex_complex(plus_plus_complex(K),B2) )
     => ( aa_complex_complex(minus_minus_complex(A),B3) = aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,K)),aa_complex_complex(minus_minus_complex(A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_3960_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B2: code_integer,A: code_integer] :
      ( ( B3 = plus_p5714425477246183910nteger(K,B2) )
     => ( minus_8373710615458151222nteger(A,B3) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K),minus_8373710615458151222nteger(A,B2)) ) ) ).

% group_cancel.sub2
tff(fact_3961_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B2: rat,A: rat] :
      ( ( B3 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B2) )
     => ( aa_rat_rat(minus_minus_rat(A),B3) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,K)),aa_rat_rat(minus_minus_rat(A),B2)) ) ) ).

% group_cancel.sub2
tff(fact_3962_dvd__div__neg,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(uminus_uminus_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) ) ) ).

% dvd_div_neg
tff(fact_3963_dvd__div__neg,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B2)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% dvd_div_neg
tff(fact_3964_dvd__div__neg,axiom,
    ! [B2: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(A),aa_complex_complex(uminus1482373934393186551omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) ) ) ).

% dvd_div_neg
tff(fact_3965_dvd__div__neg,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B2)) ) ) ).

% dvd_div_neg
tff(fact_3966_dvd__div__neg,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
     => ( aa_rat_rat(divide_divide_rat(A),aa_rat_rat(uminus_uminus_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) ) ) ).

% dvd_div_neg
tff(fact_3967_dvd__neg__div,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,A)),B2) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) ) ) ).

% dvd_neg_div
tff(fact_3968_dvd__neg__div,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B2) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)) ) ) ).

% dvd_neg_div
tff(fact_3969_dvd__neg__div,axiom,
    ! [B2: complex,A: complex] :
      ( aa_complex_o(dvd_dvd_complex(B2),A)
     => ( aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),B2) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) ) ) ).

% dvd_neg_div
tff(fact_3970_dvd__neg__div,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,divide6298287555418463151nteger(A,B2)) ) ) ).

% dvd_neg_div
tff(fact_3971_dvd__neg__div,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,B2),A)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(uminus_uminus_rat,A)),B2) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) ) ) ).

% dvd_neg_div
tff(fact_3972_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,U),U))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_3973_pos__zmult__eq__1__iff__lemma,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
     => ( ( Mb = one_one_int )
        | ( Mb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_3974_zmult__eq__1__iff,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb) = one_one_int )
    <=> ( ( ( Mb = one_one_int )
          & ( Nb = one_one_int ) )
        | ( ( Mb = aa_int_int(uminus_uminus_int,one_one_int) )
          & ( Nb = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_3975_minus__int__code_I2_J,axiom,
    ! [L: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),L) = aa_int_int(uminus_uminus_int,L) ).

% minus_int_code(2)
tff(fact_3976_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_int_int(uminus_uminus_int,L)) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
tff(fact_3977_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
tff(fact_3978_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(uminus_uminus_real,Y)) ).

% minus_real_def
tff(fact_3979_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_3980_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_3981_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_3982_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_3983_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_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_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_3991_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_3992_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_3993_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_3994_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_3995_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_3996_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_3997_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_3998_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_3999_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_4000_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_4001_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_4002_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_4003_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_4004_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_4005_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_4006_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_4007_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_4008_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_4009_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_4010_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_4011_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_le_neg_numeral
tff(fact_4012_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_le_neg_numeral
tff(fact_4013_not__one__le__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_le_neg_numeral
tff(fact_4014_not__one__le__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_le_neg_numeral
tff(fact_4015_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_le_neg_one
tff(fact_4016_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_4017_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_4018_not__numeral__le__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_4019_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),aa_real_real(uminus_uminus_real,one_one_real)) ).

% neg_numeral_le_neg_one
tff(fact_4020_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_4021_neg__numeral__le__neg__one,axiom,
    ! [Mb: 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,Mb))),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_4022_neg__numeral__le__neg__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_4023_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_le_numeral
tff(fact_4024_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_le_numeral
tff(fact_4025_neg__one__le__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_le_numeral
tff(fact_4026_neg__one__le__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_le_numeral
tff(fact_4027_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_le_one
tff(fact_4028_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_4029_neg__numeral__le__one,axiom,
    ! [Mb: 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,Mb))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_4030_neg__numeral__le__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_le_one
tff(fact_4031_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4032_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4033_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4034_not__neg__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_neg_one_less_neg_numeral
tff(fact_4035_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))) ).

% not_one_less_neg_numeral
tff(fact_4036_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))) ).

% not_one_less_neg_numeral
tff(fact_4037_not__one__less__neg__numeral,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))) ).

% not_one_less_neg_numeral
tff(fact_4038_not__one__less__neg__numeral,axiom,
    ! [Mb: 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,Mb))) ).

% not_one_less_neg_numeral
tff(fact_4039_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) ).

% not_numeral_less_neg_one
tff(fact_4040_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_4041_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_4042_not__numeral__less__neg__one,axiom,
    ! [Mb: num] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_4043_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_num_real(numeral_numeral_real,Mb)) ).

% neg_one_less_numeral
tff(fact_4044_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Mb)) ).

% neg_one_less_numeral
tff(fact_4045_neg__one__less__numeral,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(Mb)) ).

% neg_one_less_numeral
tff(fact_4046_neg__one__less__numeral,axiom,
    ! [Mb: 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,Mb)) ).

% neg_one_less_numeral
tff(fact_4047_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),one_one_real) ).

% neg_numeral_less_one
tff(fact_4048_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int) ).

% neg_numeral_less_one
tff(fact_4049_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_4050_neg__numeral__less__one,axiom,
    ! [Mb: num] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_4051_eq__minus__divide__eq,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C)) )
    <=> $ite(C != zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C) = aa_real_real(uminus_uminus_real,B2),A = zero_zero_real) ) ).

% eq_minus_divide_eq
tff(fact_4052_eq__minus__divide__eq,axiom,
    ! [A: complex,B2: complex,C: complex] :
      ( ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C)) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(A),C) = aa_complex_complex(uminus1482373934393186551omplex,B2),A = zero_zero_complex) ) ).

% eq_minus_divide_eq
tff(fact_4053_eq__minus__divide__eq,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(uminus_uminus_rat,B2),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_4054_minus__divide__eq__eq,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C)) = A )
    <=> $ite(C != zero_zero_real,aa_real_real(uminus_uminus_real,B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C),A = zero_zero_real) ) ).

% minus_divide_eq_eq
tff(fact_4055_minus__divide__eq__eq,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(B2),C)) = A )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2) = aa_complex_complex(times_times_complex(A),C),A = zero_zero_complex) ) ).

% minus_divide_eq_eq
tff(fact_4056_minus__divide__eq__eq,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(B2),C)) = A )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_4057_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: real,A: real,C: real] :
      ( ( B2 != zero_zero_real )
     => ( ( aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = C )
      <=> ( aa_real_real(uminus_uminus_real,A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4058_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: complex,A: complex,C: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = C )
      <=> ( aa_complex_complex(uminus1482373934393186551omplex,A) = aa_complex_complex(times_times_complex(C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4059_nonzero__neg__divide__eq__eq,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) = C )
      <=> ( aa_rat_rat(uminus_uminus_rat,A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_4060_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: real,C: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( ( C = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) )
      <=> ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),B2) = aa_real_real(uminus_uminus_real,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4061_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: complex,C: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( ( C = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) )
      <=> ( aa_complex_complex(times_times_complex(C),B2) = aa_complex_complex(uminus1482373934393186551omplex,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4062_nonzero__neg__divide__eq__eq2,axiom,
    ! [B2: rat,C: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( ( C = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),B2)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B2) = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_4063_mult__1s__ring__1_I2_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B2),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(2)
tff(fact_4064_mult__1s__ring__1_I2_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B2),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(2)
tff(fact_4065_mult__1s__ring__1_I2_J,axiom,
    ! [B2: complex] : aa_complex_complex(times_times_complex(B2),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(2)
tff(fact_4066_mult__1s__ring__1_I2_J,axiom,
    ! [B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% mult_1s_ring_1(2)
tff(fact_4067_mult__1s__ring__1_I2_J,axiom,
    ! [B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))) = aa_rat_rat(uminus_uminus_rat,B2) ).

% mult_1s_ring_1(2)
tff(fact_4068_mult__1s__ring__1_I1_J,axiom,
    ! [B2: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,one))),B2) = aa_real_real(uminus_uminus_real,B2) ).

% mult_1s_ring_1(1)
tff(fact_4069_mult__1s__ring__1_I1_J,axiom,
    ! [B2: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,one))),B2) = aa_int_int(uminus_uminus_int,B2) ).

% mult_1s_ring_1(1)
tff(fact_4070_mult__1s__ring__1_I1_J,axiom,
    ! [B2: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one))),B2) = aa_complex_complex(uminus1482373934393186551omplex,B2) ).

% mult_1s_ring_1(1)
tff(fact_4071_mult__1s__ring__1_I1_J,axiom,
    ! [B2: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ).

% mult_1s_ring_1(1)
tff(fact_4072_mult__1s__ring__1_I1_J,axiom,
    ! [B2: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,one))),B2) = aa_rat_rat(uminus_uminus_rat,B2) ).

% mult_1s_ring_1(1)
tff(fact_4073_divide__eq__minus__1__iff,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> ( ( B2 != zero_zero_real )
        & ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4074_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( B2 != zero_zero_complex )
        & ( A = aa_complex_complex(uminus1482373934393186551omplex,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4075_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(divide_divide_rat(A),B2) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> ( ( B2 != zero_zero_rat )
        & ( A = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_4076_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_4077_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_4078_uminus__numeral__One,axiom,
    aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(one)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% uminus_numeral_One
tff(fact_4079_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_4080_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_4081_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_4082_power__minus,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(power_power_int(A),Nb)) ).

% power_minus
tff(fact_4083_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_4084_power__minus,axiom,
    ! [A: code_integer,Nb: nat] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)),power_8256067586552552935nteger(A,Nb)) ).

% power_minus
tff(fact_4085_power__minus,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,A)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(power_power_rat(A),Nb)) ).

% power_minus
tff(fact_4086_power__minus__Bit0,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4087_power__minus__Bit0,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4088_power__minus__Bit0,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4089_power__minus__Bit0,axiom,
    ! [Xb: code_integer,K: num] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4090_power__minus__Bit0,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) = aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% power_minus_Bit0
tff(fact_4091_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_less_0_iff
tff(fact_4092_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_4093_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_4094_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(uminus_uminus_real,Xb)) ) ).

% real_add_le_0_iff
tff(fact_4095_zmod__zminus2__eq__if,axiom,
    ! [A: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_4096_zmod__zminus1__eq__if,axiom,
    ! [A: int,B2: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,A)),B2) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2))) ).

% zmod_zminus1_eq_if
tff(fact_4097_pos__minus__divide__less__eq,axiom,
    ! [C: real,B2: 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,B2),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4098_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B2: 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(B2),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4099_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),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,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4100_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4101_neg__minus__divide__less__eq,axiom,
    ! [C: real,B2: 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,B2),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,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4102_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B2: 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(B2),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4103_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4104_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_4105_minus__divide__less__eq,axiom,
    ! [B2: 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,B2),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,B2)),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,B2)),aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4106_minus__divide__less__eq,axiom,
    ! [B2: 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(B2),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,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_4107_less__minus__divide__eq,axiom,
    ! [A: real,B2: 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,B2),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,B2)),
          $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,B2)),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_4108_less__minus__divide__eq,axiom,
    ! [A: rat,B2: 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(B2),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)),
          $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,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_4109_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: real,C: real,W: num] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),C) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
    <=> $ite(C != zero_zero_real,B2 = 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_4110_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: complex,C: complex,W: num] :
      ( ( aa_complex_complex(divide1717551699836669952omplex(B2),C) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> $ite(C != zero_zero_complex,B2 = 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_4111_divide__eq__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( ( aa_rat_rat(divide_divide_rat(B2),C) = aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) )
    <=> $ite(C != zero_zero_rat,B2 = aa_rat_rat(aa_rat_fun_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_4112_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: 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,B2),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) = B2,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) = zero_zero_real) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4113_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: complex,C: complex] :
      ( ( aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(B2),C) )
    <=> $ite(C != zero_zero_complex,aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))),C) = B2,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) = zero_zero_complex) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4114_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: rat,C: rat] :
      ( ( aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(B2),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C) = B2,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4115_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: real,Zb: real,B2: 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),Zb))),B2) = $ite(Zb = zero_zero_real,B2,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,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(3)
tff(fact_4116_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: complex,Zb: complex,B2: complex] :
      aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Zb))),B2) = $ite(Zb = zero_zero_complex,B2,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(3)
tff(fact_4117_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Zb: rat,B2: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Zb))),B2) = $ite(Zb = zero_zero_rat,B2,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(3)
tff(fact_4118_minus__divide__add__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),Zb) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4119_minus__divide__add__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Zb))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Zb))),Zb) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4120_minus__divide__add__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Zb))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),Zb) ) ) ).

% minus_divide_add_eq_iff
tff(fact_4121_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: real,Zb: real,B2: 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),Zb))),B2) = $ite(Zb = zero_zero_real,aa_real_real(uminus_uminus_real,B2),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,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(6)
tff(fact_4122_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: complex,Zb: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(A),Zb))),B2) = $ite(Zb = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),aa_complex_complex(times_times_complex(B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(6)
tff(fact_4123_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Zb: rat,B2: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(A),Zb))),B2) = $ite(Zb = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(6)
tff(fact_4124_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: real,Zb: real,B2: real] :
      aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),Zb)),B2) = $ite(Zb = zero_zero_real,aa_real_real(uminus_uminus_real,B2),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,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(5)
tff(fact_4125_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: complex,Zb: complex,B2: complex] :
      aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(A),Zb)),B2) = $ite(Zb = zero_zero_complex,aa_complex_complex(uminus1482373934393186551omplex,B2),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(A),aa_complex_complex(times_times_complex(B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(5)
tff(fact_4126_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Zb: rat,B2: rat] :
      aa_rat_rat(minus_minus_rat(aa_rat_rat(divide_divide_rat(A),Zb)),B2) = $ite(Zb = zero_zero_rat,aa_rat_rat(uminus_uminus_rat,B2),aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),Zb))),Zb)) ).

% add_divide_eq_if_simps(5)
tff(fact_4127_minus__divide__diff__eq__iff,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( ( Zb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Zb))),Y) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))),Zb) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4128_minus__divide__diff__eq__iff,axiom,
    ! [Zb: complex,Xb: complex,Y: complex] :
      ( ( Zb != zero_zero_complex )
     => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(divide1717551699836669952omplex(Xb),Zb))),Y) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_complex_complex(times_times_complex(Y),Zb))),Zb) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4129_minus__divide__diff__eq__iff,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( ( Zb != zero_zero_rat )
     => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_rat_rat(divide_divide_rat(Xb),Zb))),Y) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))),Zb) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_4130_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_4131_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_4132_power2__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4133_power2__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4134_power2__eq__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_complex(power_power_complex(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_complex_complex(uminus1482373934393186551omplex,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4135_power2__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4136_power2__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% power2_eq_iff
tff(fact_4137_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A3),B3)
     => ( 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(aa_int_fun_int_int(divide_divide_int,B3),Nb)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A3),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4138_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_4139_pos__minus__divide__le__eq,axiom,
    ! [C: real,B2: 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,B2),C))),A)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4140_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B2: 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(B2),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4141_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),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,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4142_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4143_neg__minus__divide__le__eq,axiom,
    ! [C: real,B2: 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,B2),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,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4144_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B2: 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(B2),C))),A)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4145_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B2: 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,B2),C)))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4146_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B2: 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(B2),C)))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4147_minus__divide__le__eq,axiom,
    ! [B2: 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,B2),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,B2)),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,B2)),aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4148_minus__divide__le__eq,axiom,
    ! [B2: 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(B2),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,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),zero_zero_rat),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_4149_le__minus__divide__eq,axiom,
    ! [A: real,B2: 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,B2),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,B2)),
          $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,B2)),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_4150_le__minus__divide__eq,axiom,
    ! [A: rat,B2: 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(B2),C)))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(uminus_uminus_rat,B2)),
          $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,B2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_4151_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: 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,B2),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)),B2),
          $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,B2),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_4152_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: 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(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_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_4153_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),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_4154_divide__less__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B2),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_4155_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_4156_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_4157_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_4158_power2__eq__1__iff,axiom,
    ! [A: 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_4159_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_4160_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_4161_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_4162_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_4163_uminus__power__if,axiom,
    ! [A: code_integer,Nb: nat] :
      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),power_8256067586552552935nteger(A,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(A,Nb))) ).

% uminus_power_if
tff(fact_4164_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_4165_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_4166_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_4167_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_4168_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K)) = 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_4169_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_4170_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,U),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% realpow_square_minus_le
tff(fact_4171_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),L)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,K)),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),one_one_int)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_4172_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B2) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),one_one_int) ) ) ).

% zmod_minus1
tff(fact_4173_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B2) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_4174_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A: int] :
      ( ( B2 != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B2)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B2) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_4175_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: 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,B2),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)),B2),
          $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,B2),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_4176_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B2: 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(B2),C))
    <=> $ite(
          aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),C),
          aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B2),
          $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,B2),aa_rat_rat(aa_rat_fun_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_4177_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: 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,B2),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,B2),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)),B2),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_4178_divide__le__eq__numeral_I2_J,axiom,
    ! [B2: rat,C: rat,W: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(B2),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,B2),aa_rat_rat(aa_rat_fun_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(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))),C)),B2),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_4179_square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ) ) ).

% square_le_1
tff(fact_4180_square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Xb)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),one_one_Code_integer)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_4181_square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_4182_square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_4183_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_4184_minus__power__mult__self,axiom,
    ! [A: int,Nb: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),Nb)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% minus_power_mult_self
tff(fact_4185_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_4186_minus__power__mult__self,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb)) = 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_4187_minus__power__mult__self,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(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_4188_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_4189_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_4190_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_4191_minus__one__power__iff,axiom,
    ! [Nb: nat] :
      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_4192_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_4193_minus__1__div__exp__eq__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_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_4194_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)),zero_zero_int)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% div_pos_neg_trivial
tff(fact_4195_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_4196_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_4197_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_4198_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_4199_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_4200_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_4201_power__minus1__odd,axiom,
    ! [Nb: nat] : 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_4202_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_4203_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ) )
           => aa_int_o(P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_4204_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_4205_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_4206_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(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb)))),aa_int_int(bit_ri631733984087533419it_int(Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4207_mult__less__iff1,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Zb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Zb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4208_mult__less__iff1,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Zb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Zb))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4209_mult__less__iff1,axiom,
    ! [Zb: int,Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Zb)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Zb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ) ).

% mult_less_iff1
tff(fact_4210_compl__le__compl__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb),uminus5710092332889474511et_nat(Y))
    <=> ord_less_eq_set_nat(Y,Xb) ) ).

% compl_le_compl_iff
tff(fact_4211_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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_4212_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_concat_bit(Nb,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),L))) ).

% concat_bit_Suc
tff(fact_4213_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_4214_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4215_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% flip_bit_0
tff(fact_4216_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member_nat(zero_zero_nat,aa_nat_set_nat(nat_set_decode,Xb))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Xb) ) ).

% set_decode_0
tff(fact_4217_semiring__norm_I90_J,axiom,
    ! [Mb: num,Nb: num] :
      ( ( aa_num_num(bit1,Mb) = aa_num_num(bit1,Nb) )
    <=> ( Mb = Nb ) ) ).

% semiring_norm(90)
tff(fact_4218_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_4219_semiring__norm_I88_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit0,Mb) != aa_num_num(bit1,Nb) ).

% semiring_norm(88)
tff(fact_4220_semiring__norm_I89_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(bit1,Mb) != aa_num_num(bit0,Nb) ).

% semiring_norm(89)
tff(fact_4221_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one != aa_num_num(bit1,Nb) ).

% semiring_norm(84)
tff(fact_4222_semiring__norm_I86_J,axiom,
    ! [Mb: num] : aa_num_num(bit1,Mb) != one ).

% semiring_norm(86)
tff(fact_4223_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4224_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4225_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4226_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),zero_n356916108424825756nteger((Q)))
    <=> ( (P)
       => (Q) ) ) ).

% of_bool_less_eq_iff
tff(fact_4227_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = zero_zero_real )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4228_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = zero_zero_rat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4229_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = zero_zero_nat )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4230_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_4231_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = zero_z3403309356797280102nteger )
    <=> ~ (P) ) ).

% of_bool_eq_0_iff
tff(fact_4232_of__bool__eq_I1_J,axiom,
    zero_n3304061248610475627l_real($false) = zero_zero_real ).

% of_bool_eq(1)
tff(fact_4233_of__bool__eq_I1_J,axiom,
    zero_n2052037380579107095ol_rat($false) = zero_zero_rat ).

% of_bool_eq(1)
tff(fact_4234_of__bool__eq_I1_J,axiom,
    zero_n2687167440665602831ol_nat($false) = zero_zero_nat ).

% of_bool_eq(1)
tff(fact_4235_of__bool__eq_I1_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$false) = zero_zero_int ).

% of_bool_eq(1)
tff(fact_4236_of__bool__eq_I1_J,axiom,
    zero_n356916108424825756nteger($false) = zero_z3403309356797280102nteger ).

% of_bool_eq(1)
tff(fact_4237_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4238_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4239_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q)))
    <=> ( ~ (P)
        & (Q) ) ) ).

% of_bool_less_iff
tff(fact_4240_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_4241_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_4242_of__bool__eq_I2_J,axiom,
    zero_n1201886186963655149omplex($true) = one_one_complex ).

% of_bool_eq(2)
tff(fact_4243_of__bool__eq_I2_J,axiom,
    zero_n3304061248610475627l_real($true) = one_one_real ).

% of_bool_eq(2)
tff(fact_4244_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4245_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat($true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4246_of__bool__eq_I2_J,axiom,
    aa_o_int(zero_n2684676970156552555ol_int,$true) = one_one_int ).

% of_bool_eq(2)
tff(fact_4247_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_4248_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n1201886186963655149omplex((P)) = one_one_complex )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4249_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n3304061248610475627l_real((P)) = one_one_real )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4250_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat((P)) = one_one_rat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4251_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2687167440665602831ol_nat((P)) = one_one_nat )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4252_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P)) = one_one_int )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4253_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger((P)) = one_one_Code_integer )
    <=> (P) ) ).

% of_bool_eq_1_iff
tff(fact_4254_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_4255_semiring__norm_I73_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(73)
tff(fact_4256_semiring__norm_I80_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(80)
tff(fact_4257_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),zero_n3304061248610475627l_real((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4258_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4259_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P)))
    <=> (P) ) ).

% zero_less_of_bool_iff
tff(fact_4260_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_4261_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_4262_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_n3304061248610475627l_real((P))),one_one_real)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4263_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4264_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat)
    <=> ~ (P) ) ).

% of_bool_less_one_iff
tff(fact_4265_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_4266_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_4267_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n1201886186963655149omplex(~ (P)) = aa_complex_complex(minus_minus_complex(one_one_complex),zero_n1201886186963655149omplex((P))) ).

% of_bool_not_iff
tff(fact_4268_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n3304061248610475627l_real(~ (P)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),zero_n3304061248610475627l_real((P))) ).

% of_bool_not_iff
tff(fact_4269_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n2052037380579107095ol_rat(~ (P)) = aa_rat_rat(minus_minus_rat(one_one_rat),zero_n2052037380579107095ol_rat((P))) ).

% of_bool_not_iff
tff(fact_4270_of__bool__not__iff,axiom,
    ! [P: $o] : aa_o_int(zero_n2684676970156552555ol_int,~ (P)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),aa_o_int(zero_n2684676970156552555ol_int,(P))) ).

% of_bool_not_iff
tff(fact_4271_of__bool__not__iff,axiom,
    ! [P: $o] : zero_n356916108424825756nteger(~ (P)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((P))) ).

% of_bool_not_iff
tff(fact_4272_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = zero_n2687167440665602831ol_nat(Nb != aa_nat_nat(suc,zero_zero_nat)) ).

% Suc_0_mod_eq
tff(fact_4273_semiring__norm_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(9)
tff(fact_4274_semiring__norm_I7_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(Mb),Nb)) ).

% semiring_norm(7)
tff(fact_4275_semiring__norm_I15_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(aa_num_num(bit1,Mb)),Nb)) ).

% semiring_norm(15)
tff(fact_4276_semiring__norm_I14_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(times_times_num(Mb),aa_num_num(bit1,Nb))) ).

% semiring_norm(14)
tff(fact_4277_semiring__norm_I72_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(72)
tff(fact_4278_semiring__norm_I81_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(81)
tff(fact_4279_semiring__norm_I70_J,axiom,
    ! [Mb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),one) ).

% semiring_norm(70)
tff(fact_4280_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_4281_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_4282_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_4283_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,W)) ).

% zdiv_numeral_Bit1
tff(fact_4284_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit0,Nb)) = aa_num_num(bit1,Nb) ).

% semiring_norm(3)
tff(fact_4285_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(Nb),one)) ).

% semiring_norm(4)
tff(fact_4286_semiring__norm_I5_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit0,Mb)),one) = aa_num_num(bit1,Mb) ).

% semiring_norm(5)
tff(fact_4287_semiring__norm_I8_J,axiom,
    ! [Mb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),one) = aa_num_num(bit0,aa_num_num(plus_plus_num(Mb),one)) ).

% semiring_norm(8)
tff(fact_4288_semiring__norm_I10_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit0,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),one)) ).

% semiring_norm(10)
tff(fact_4289_semiring__norm_I16_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_num(times_times_num(aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = aa_num_num(bit1,aa_num_num(plus_plus_num(aa_num_num(plus_plus_num(Mb),Nb)),aa_num_num(bit0,aa_num_num(times_times_num(Mb),Nb)))) ).

% semiring_norm(16)
tff(fact_4290_semiring__norm_I74_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb) ) ).

% semiring_norm(74)
tff(fact_4291_semiring__norm_I79_J,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb))
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb) ) ).

% semiring_norm(79)
tff(fact_4292_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),zero_n2687167440665602831ol_nat((P2)))
    <=> (P2) ) ).

% odd_of_bool_self
tff(fact_4293_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_4294_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_4295_of__bool__half__eq__0,axiom,
    ! [B2: $o] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,zero_n2687167440665602831ol_nat((B2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = zero_zero_nat ).

% of_bool_half_eq_0
tff(fact_4296_of__bool__half__eq__0,axiom,
    ! [B2: $o] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_o_int(zero_n2684676970156552555ol_int,(B2))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int ).

% of_bool_half_eq_0
tff(fact_4297_of__bool__half__eq__0,axiom,
    ! [B2: $o] : divide6298287555418463151nteger(zero_n356916108424825756nteger((B2)),numera6620942414471956472nteger(aa_num_num(bit0,one))) = zero_z3403309356797280102nteger ).

% of_bool_half_eq_0
tff(fact_4298_Suc__div__eq__add3__div__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_4299_div__Suc__eq__div__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_4300_mod__Suc__eq__mod__add3,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Nb)))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_4301_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [Mb: nat,V: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),aa_num_nat(numeral_numeral_nat,V)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),aa_num_nat(numeral_numeral_nat,V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_4302_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( member_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(nat_set_decode,Xb))
    <=> member_nat(Nb,aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% set_decode_Suc
tff(fact_4303_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4304_bits__1__div__exp,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4305_bits__1__div__exp,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4306_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4307_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_o_int(zero_n2684676970156552555ol_int,Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4308_one__div__2__pow__eq,axiom,
    ! [Nb: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = zero_n356916108424825756nteger(Nb = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4309_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,V))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(aa_int_fun_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_4310_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% one_mod_2_pow_eq
tff(fact_4311_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),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_4312_one__mod__2__pow__eq,axiom,
    ! [Nb: nat] : modulo364778990260209775nteger(one_one_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_4313_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(Nb),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4314_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n2687167440665602831ol_nat((P2)) = zero_n2687167440665602831ol_nat((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4315_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( aa_o_int(zero_n2684676970156552555ol_int,(P2)) = aa_o_int(zero_n2684676970156552555ol_int,(Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4316_of__bool__eq__iff,axiom,
    ! [P2: $o,Q2: $o] :
      ( ( zero_n356916108424825756nteger((P2)) = zero_n356916108424825756nteger((Q2)) )
    <=> ( (P2)
      <=> (Q2) ) ) ).

% of_bool_eq_iff
tff(fact_4317_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n3304061248610475627l_real(( (P)
        & (Q) )) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,zero_n3304061248610475627l_real((P))),zero_n3304061248610475627l_real((Q))) ).

% of_bool_conj
tff(fact_4318_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2052037380579107095ol_rat(( (P)
        & (Q) )) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_n2052037380579107095ol_rat((P))),zero_n2052037380579107095ol_rat((Q))) ).

% of_bool_conj
tff(fact_4319_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      zero_n2687167440665602831ol_nat(( (P)
        & (Q) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat((P))),zero_n2687167440665602831ol_nat((Q))) ).

% of_bool_conj
tff(fact_4320_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      aa_o_int(zero_n2684676970156552555ol_int,
        ( (P)
        & (Q) )) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),aa_o_int(zero_n2684676970156552555ol_int,(Q))) ).

% of_bool_conj
tff(fact_4321_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_4322_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_4323_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one != aa_num_num(bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_4324_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,Mb: nat,L: int,R2: int] : aa_int_int(bit_concat_bit(Nb,K),aa_int_int(bit_concat_bit(Mb,L),R2)) = aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb),aa_int_int(bit_concat_bit(Nb,K),L)),R2) ).

% concat_bit_assoc
tff(fact_4325_subset__decode__imp__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ord_less_eq_set_nat(aa_nat_set_nat(nat_set_decode,Mb),aa_nat_set_nat(nat_set_decode,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% subset_decode_imp_le
tff(fact_4326_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),zero_n3304061248610475627l_real((P))) ).

% zero_less_eq_of_bool
tff(fact_4327_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),zero_n2052037380579107095ol_rat((P))) ).

% zero_less_eq_of_bool
tff(fact_4328_zero__less__eq__of__bool,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),zero_n2687167440665602831ol_nat((P))) ).

% zero_less_eq_of_bool
tff(fact_4329_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_4330_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_4331_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_n3304061248610475627l_real((P))),one_one_real) ).

% of_bool_less_eq_one
tff(fact_4332_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_n2052037380579107095ol_rat((P))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_4333_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((P))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_4334_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_o_int(zero_n2684676970156552555ol_int,(P))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_4335_of__bool__less__eq__one,axiom,
    ! [P: $o] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_n356916108424825756nteger((P))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_4336_split__of__bool__asm,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_complex_o(P,one_one_complex) )
          | ( ~ (P2)
            & ~ aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool_asm
tff(fact_4337_split__of__bool__asm,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_real_o(P,one_one_real) )
          | ( ~ (P2)
            & ~ aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool_asm
tff(fact_4338_split__of__bool__asm,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_rat_o(P,one_one_rat) )
          | ( ~ (P2)
            & ~ aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4339_split__of__bool__asm,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ~ ( ( (P2)
            & ~ aa_nat_o(P,one_one_nat) )
          | ( ~ (P2)
            & ~ aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4340_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_4341_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_4342_split__of__bool,axiom,
    ! [P: fun_complex_o,P2: $o] :
      ( aa_complex_o(P,zero_n1201886186963655149omplex((P2)))
    <=> ( ( (P2)
         => aa_complex_o(P,one_one_complex) )
        & ( ~ (P2)
         => aa_complex_o(P,zero_zero_complex) ) ) ) ).

% split_of_bool
tff(fact_4343_split__of__bool,axiom,
    ! [P: fun_real_o,P2: $o] :
      ( aa_real_o(P,zero_n3304061248610475627l_real((P2)))
    <=> ( ( (P2)
         => aa_real_o(P,one_one_real) )
        & ( ~ (P2)
         => aa_real_o(P,zero_zero_real) ) ) ) ).

% split_of_bool
tff(fact_4344_split__of__bool,axiom,
    ! [P: fun_rat_o,P2: $o] :
      ( aa_rat_o(P,zero_n2052037380579107095ol_rat((P2)))
    <=> ( ( (P2)
         => aa_rat_o(P,one_one_rat) )
        & ( ~ (P2)
         => aa_rat_o(P,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4345_split__of__bool,axiom,
    ! [P: fun_nat_o,P2: $o] :
      ( aa_nat_o(P,zero_n2687167440665602831ol_nat((P2)))
    <=> ( ( (P2)
         => aa_nat_o(P,one_one_nat) )
        & ( ~ (P2)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4346_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_4347_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_4348_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n1201886186963655149omplex((P2)) = $ite((P2),one_one_complex,zero_zero_complex) ).

% of_bool_def
tff(fact_4349_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n3304061248610475627l_real((P2)) = $ite((P2),one_one_real,zero_zero_real) ).

% of_bool_def
tff(fact_4350_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2052037380579107095ol_rat((P2)) = $ite((P2),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4351_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n2687167440665602831ol_nat((P2)) = $ite((P2),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4352_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_4353_of__bool__def,axiom,
    ! [P2: $o] :
      zero_n356916108424825756nteger((P2)) = $ite((P2),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_4354_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] : Y != aa_num_num(bit0,X22)
       => ~ ! [X33: num] : Y != aa_num_num(bit1,X33) ) ) ).

% num.exhaust
tff(fact_4355_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_4356_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_4357_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(bit1,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Nb)),aa_num_rat(numeral_numeral_rat,Nb))),one_one_rat) ).

% numeral_Bit1
tff(fact_4358_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_4359_numeral__Bit1,axiom,
    ! [Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Nb))),one_one_int) ).

% numeral_Bit1
tff(fact_4360_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_4361_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: 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,Mb))),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_4362_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_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_4363_cong__exp__iff__simps_I10_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit0,Mb)),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_4364_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: 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,Mb))),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_4365_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) != aa_int_int(aa_int_fun_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_4366_cong__exp__iff__simps_I12_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] : modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Mb)),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_4367_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: 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,Mb))),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,Mb)),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_4368_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = aa_int_int(aa_int_fun_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_4369_cong__exp__iff__simps_I13_J,axiom,
    ! [Mb: num,Q2: num,Nb: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Nb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Q2)) = modulo364778990260209775nteger(numera6620942414471956472nteger(Nb),numera6620942414471956472nteger(Q2)) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_4370_power__minus__Bit1,axiom,
    ! [Xb: real,K: num] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_real_real(uminus_uminus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4371_power__minus__Bit1,axiom,
    ! [Xb: int,K: num] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4372_power__minus__Bit1,axiom,
    ! [Xb: complex,K: num] : aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4373_power__minus__Bit1,axiom,
    ! [Xb: code_integer,K: num] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4374_power__minus__Bit1,axiom,
    ! [Xb: rat,K: num] : aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K))) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,K)))) ).

% power_minus_Bit1
tff(fact_4375_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_4376_numeral__Bit1__div__2,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_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_4377_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_4378_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_4379_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_4380_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_4381_cong__exp__iff__simps_I3_J,axiom,
    ! [Nb: num,Q2: num] : aa_int_int(aa_int_fun_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_4382_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_4383_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_4384_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_4385_power3__eq__cube,axiom,
    ! [A: rat] : aa_nat_rat(power_power_rat(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),A) ).

% power3_eq_cube
tff(fact_4386_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_4387_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_4388_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_4389_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_4390_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] : zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% of_bool_odd_eq_mod_2
tff(fact_4391_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(aa_int_fun_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_4392_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_4393_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_4394_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,Nb: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Nb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
tff(fact_4395_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_4396_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,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,Mb)),aa_num_nat(numeral_numeral_nat,Q2)) = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4397_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,one)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Q2))) )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Q2)) = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4398_cong__exp__iff__simps_I11_J,axiom,
    ! [Mb: num,Q2: num] :
      ( ( modulo364778990260209775nteger(numera6620942414471956472nteger(aa_num_num(bit1,Mb)),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) = modulo364778990260209775nteger(numera6620942414471956472nteger(one),numera6620942414471956472nteger(aa_num_num(bit0,Q2))) )
    <=> ( modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Q2)) = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
tff(fact_4399_Suc__div__eq__add3__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_4400_Suc__mod__eq__add3__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(suc,aa_nat_nat(suc,aa_nat_nat(suc,Mb)))),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))),Mb)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_4401_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,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,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_4402_bits__induct,axiom,
    ! [P: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,(B4))),aa_int_int(aa_int_fun_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_4403_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_4404_compl__le__swap2,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y),Xb)
     => ord_less_eq_set_nat(uminus5710092332889474511et_nat(Xb),Y) ) ).

% compl_le_swap2
tff(fact_4405_compl__le__swap1,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( ord_less_eq_set_nat(Y,uminus5710092332889474511et_nat(Xb))
     => ord_less_eq_set_nat(Xb,uminus5710092332889474511et_nat(Y)) ) ).

% compl_le_swap1
tff(fact_4406_compl__mono,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ord_less_eq_set_nat(uminus5710092332889474511et_nat(Y),uminus5710092332889474511et_nat(Xb)) ) ).

% compl_mono
tff(fact_4407_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4408_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)) ).

% exp_mod_exp
tff(fact_4409_exp__mod__exp,axiom,
    ! [Mb: nat,Nb: nat] : modulo364778990260209775nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb)) ).

% exp_mod_exp
tff(fact_4410_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4411_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Mb) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4412_exp__div__exp__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      divide6298287555418463151nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Mb) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb) ))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ).

% exp_div_exp_eq
tff(fact_4413_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_4414_mod__exhaust__less__4,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = zero_zero_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = one_one_nat )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) )
      | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,aa_num_num(bit0,one)))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one)) ) ) ).

% mod_exhaust_less_4
tff(fact_4415_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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_4416_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_4417_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_4418_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_4419_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_4420_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_4421_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_4422_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_4423_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_4424_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_4425_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_4426_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B2: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B2 )
          & ( 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,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4427_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B2: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4428_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B2: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B2 )
          & ( 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,B2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4429_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B2: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B2 )
          & ( C != D ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4430_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_4431_dbl__dec__simps_I3_J,axiom,
    aa_complex_complex(neg_nu6511756317524482435omplex,one_one_complex) = one_one_complex ).

% dbl_dec_simps(3)
tff(fact_4432_dbl__dec__simps_I3_J,axiom,
    aa_real_real(neg_nu6075765906172075777c_real,one_one_real) = one_one_real ).

% dbl_dec_simps(3)
tff(fact_4433_dbl__dec__simps_I3_J,axiom,
    aa_rat_rat(neg_nu3179335615603231917ec_rat,one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_4434_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_4435_pred__numeral__simps_I1_J,axiom,
    aa_num_nat(pred_numeral,one) = zero_zero_nat ).

% pred_numeral_simps(1)
tff(fact_4436_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_4437_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_4438_dbl__inc__simps_I2_J,axiom,
    aa_complex_complex(neg_nu8557863876264182079omplex,zero_zero_complex) = one_one_complex ).

% dbl_inc_simps(2)
tff(fact_4439_dbl__inc__simps_I2_J,axiom,
    aa_real_real(neg_nu8295874005876285629c_real,zero_zero_real) = one_one_real ).

% dbl_inc_simps(2)
tff(fact_4440_dbl__inc__simps_I2_J,axiom,
    aa_rat_rat(neg_nu5219082963157363817nc_rat,zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_4441_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_4442_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_4443_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_4444_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_4445_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_4446_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_4447_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_4448_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_4449_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_4450_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_4451_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_4452_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_4453_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_4454_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_4455_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_4456_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_4457_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_4458_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_4459_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_4460_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_4461_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_4462_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_4463_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_4464_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_4465_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_4466_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_4467_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_4468_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_4469_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_4470_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_4471_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_4472_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_4473_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(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,K))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_bit0
tff(fact_4474_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(aa_int_fun_int_int(times_times_int,aa_int_int(bit_ri631733984087533419it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_4475_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_4476_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_4477_dbl__inc__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu8557863876264182079omplex,Xb) = aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_inc_def
tff(fact_4478_dbl__inc__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu8295874005876285629c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_inc_def
tff(fact_4479_dbl__inc__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu5219082963157363817nc_rat,Xb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb)),one_one_rat) ).

% dbl_inc_def
tff(fact_4480_dbl__inc__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu5851722552734809277nc_int,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_inc_def
tff(fact_4481_num_Osize__gen_I1_J,axiom,
    aa_num_nat(size_num,one) = zero_zero_nat ).

% num.size_gen(1)
tff(fact_4482_dbl__dec__def,axiom,
    ! [Xb: complex] : aa_complex_complex(neg_nu6511756317524482435omplex,Xb) = aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(Xb),Xb)),one_one_complex) ).

% dbl_dec_def
tff(fact_4483_dbl__dec__def,axiom,
    ! [Xb: real] : aa_real_real(neg_nu6075765906172075777c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Xb)),one_one_real) ).

% dbl_dec_def
tff(fact_4484_dbl__dec__def,axiom,
    ! [Xb: rat] : aa_rat_rat(neg_nu3179335615603231917ec_rat,Xb) = aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Xb)),one_one_rat) ).

% dbl_dec_def
tff(fact_4485_dbl__dec__def,axiom,
    ! [Xb: int] : aa_int_int(neg_nu3811975205180677377ec_int,Xb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Xb)),one_one_int) ).

% dbl_dec_def
tff(fact_4486_add__0__iff,axiom,
    ! [B2: real,A: real] :
      ( ( B2 = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),A) )
    <=> ( A = zero_zero_real ) ) ).

% add_0_iff
tff(fact_4487_add__0__iff,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B2),A) )
    <=> ( A = zero_zero_rat ) ) ).

% add_0_iff
tff(fact_4488_add__0__iff,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B2),A) )
    <=> ( A = zero_zero_nat ) ) ).

% add_0_iff
tff(fact_4489_add__0__iff,axiom,
    ! [B2: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,B2),A) )
    <=> ( A = zero_zero_int ) ) ).

% add_0_iff
tff(fact_4490_crossproduct__noteq,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ( ( A != B2 )
        & ( 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,B2),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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_4491_crossproduct__noteq,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ( ( A != B2 )
        & ( C != D ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),D)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_4492_crossproduct__noteq,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ( ( A != B2 )
        & ( 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,B2),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,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_4493_crossproduct__noteq,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ( ( A != B2 )
        & ( C != D ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ) ) ).

% crossproduct_noteq
tff(fact_4494_crossproduct__eq,axiom,
    ! [W: real,Y: real,Xb: real,Zb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Y)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,W),Zb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Zb ) ) ) ).

% crossproduct_eq
tff(fact_4495_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,Xb: rat,Zb: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Zb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Zb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Zb ) ) ) ).

% crossproduct_eq
tff(fact_4496_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,Xb: nat,Zb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Zb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Zb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Zb ) ) ) ).

% crossproduct_eq
tff(fact_4497_crossproduct__eq,axiom,
    ! [W: int,Y: int,Xb: int,Zb: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Zb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Zb)),aa_int_int(aa_int_fun_int_int(times_times_int,Xb),Y)) )
    <=> ( ( W = Xb )
        | ( Y = Zb ) ) ) ).

% crossproduct_eq
tff(fact_4498_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Zb) )
    <=> ( Y = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Zb) ) ) ).

% eq_diff_eq'
tff(fact_4499_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_4500_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_4501_mask__numeral,axiom,
    ! [Nb: num] : aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(numeral_numeral_nat,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(bit_se2000444600071755411sk_int,aa_num_nat(pred_numeral,Nb)))) ).

% mask_numeral
tff(fact_4502_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_4503_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))) ).

% take_bit_rec
tff(fact_4504_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_4505_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% and_int_unfold
tff(fact_4506_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_real_real(exp_real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_4507_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_4508_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4509_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4510_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4511_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_4512_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4513_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4514_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4515_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_4516_and_Oright__idem,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B2)),B2) = aa_int_int(bit_se725231765392027082nd_int(A),B2) ).

% and.right_idem
tff(fact_4517_and_Oright__idem,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2)),B2) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2) ).

% and.right_idem
tff(fact_4518_and_Oleft__idem,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(A),B2)) = aa_int_int(bit_se725231765392027082nd_int(A),B2) ).

% and.left_idem
tff(fact_4519_and_Oleft__idem,axiom,
    ! [A: nat,B2: 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),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2) ).

% and.left_idem
tff(fact_4520_and_Oidem,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),A) = A ).

% and.idem
tff(fact_4521_and_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),A) = A ).

% and.idem
tff(fact_4522_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,Xb) = aa_real_real(sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_4523_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_4524_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_4525_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_4526_bit_Oconj__zero__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),zero_zero_int) = zero_zero_int ).

% bit.conj_zero_right
tff(fact_4527_bit_Oconj__zero__left,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),Xb) = zero_zero_int ).

% bit.conj_zero_left
tff(fact_4528_zero__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(zero_zero_int),A) = zero_zero_int ).

% zero_and_eq
tff(fact_4529_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_4530_and__zero__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),zero_zero_int) = zero_zero_int ).

% and_zero_eq
tff(fact_4531_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_4532_real__sqrt__zero,axiom,
    aa_real_real(sqrt,zero_zero_real) = zero_zero_real ).

% real_sqrt_zero
tff(fact_4533_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
    <=> ( Xb = zero_zero_real ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_4534_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_4535_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_4536_real__sqrt__one,axiom,
    aa_real_real(sqrt,one_one_real) = one_one_real ).

% real_sqrt_one
tff(fact_4537_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sqrt,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% real_sqrt_eq_1_iff
tff(fact_4538_take__bit__and,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se725231765392027082nd_int(A),B2)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B2)) ).

% take_bit_and
tff(fact_4539_take__bit__and,axiom,
    ! [Nb: nat,A: nat,B2: 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),B2)) = 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),B2)) ).

% take_bit_and
tff(fact_4540_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_4541_take__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(zero_zero_nat),A) = zero_zero_int ).

% take_bit_0
tff(fact_4542_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_4543_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_4544_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_4545_and_Oleft__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),A) = A ).

% and.left_neutral
tff(fact_4546_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_4547_and_Oright__neutral,axiom,
    ! [A: code_integer] : bit_se3949692690581998587nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_4548_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_4549_bit_Oconj__one__right,axiom,
    ! [Xb: code_integer] : bit_se3949692690581998587nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = Xb ).

% bit.conj_one_right
tff(fact_4550_bit_Oconj__one__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),aa_int_int(uminus_uminus_int,one_one_int)) = Xb ).

% bit.conj_one_right
tff(fact_4551_of__int__eq__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Zb) = numera6690914467698888265omplex(Nb) )
    <=> ( Zb = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4552_of__int__eq__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( ( aa_int_real(ring_1_of_int_real,Zb) = aa_num_real(numeral_numeral_real,Nb) )
    <=> ( Zb = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4553_of__int__eq__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( ( aa_int_rat(ring_1_of_int_rat,Zb) = aa_num_rat(numeral_numeral_rat,Nb) )
    <=> ( Zb = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4554_of__int__eq__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( ( aa_int_int(ring_1_of_int_int,Zb) = aa_num_int(numeral_numeral_int,Nb) )
    <=> ( Zb = aa_num_int(numeral_numeral_int,Nb) ) ) ).

% of_int_eq_numeral_iff
tff(fact_4555_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_4556_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_4557_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_4558_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_4559_of__int__le__iff,axiom,
    ! [W: int,Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb) ) ).

% of_int_le_iff
tff(fact_4560_of__int__le__iff,axiom,
    ! [W: int,Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb) ) ).

% of_int_le_iff
tff(fact_4561_of__int__le__iff,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb) ) ).

% of_int_le_iff
tff(fact_4562_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_4563_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_4564_of__int__less__iff,axiom,
    ! [W: int,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ).

% of_int_less_iff
tff(fact_4565_of__int__less__iff,axiom,
    ! [W: int,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ).

% of_int_less_iff
tff(fact_4566_of__int__less__iff,axiom,
    ! [W: int,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ).

% of_int_less_iff
tff(fact_4567_of__int__1,axiom,
    aa_int_complex(ring_17405671764205052669omplex,one_one_int) = one_one_complex ).

% of_int_1
tff(fact_4568_of__int__1,axiom,
    aa_int_int(ring_1_of_int_int,one_one_int) = one_one_int ).

% of_int_1
tff(fact_4569_of__int__1,axiom,
    aa_int_real(ring_1_of_int_real,one_one_int) = one_one_real ).

% of_int_1
tff(fact_4570_of__int__1,axiom,
    aa_int_rat(ring_1_of_int_rat,one_one_int) = one_one_rat ).

% of_int_1
tff(fact_4571_of__int__eq__1__iff,axiom,
    ! [Zb: int] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Zb) = one_one_complex )
    <=> ( Zb = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4572_of__int__eq__1__iff,axiom,
    ! [Zb: int] :
      ( ( aa_int_int(ring_1_of_int_int,Zb) = one_one_int )
    <=> ( Zb = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4573_of__int__eq__1__iff,axiom,
    ! [Zb: int] :
      ( ( aa_int_real(ring_1_of_int_real,Zb) = one_one_real )
    <=> ( Zb = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4574_of__int__eq__1__iff,axiom,
    ! [Zb: int] :
      ( ( aa_int_rat(ring_1_of_int_rat,Zb) = one_one_rat )
    <=> ( Zb = one_one_int ) ) ).

% of_int_eq_1_iff
tff(fact_4575_of__int__mult,axiom,
    ! [W: int,Zb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,W),Zb)) = 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,Zb)) ).

% of_int_mult
tff(fact_4576_of__int__mult,axiom,
    ! [W: int,Zb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,W),Zb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Zb)) ).

% of_int_mult
tff(fact_4577_of__int__mult,axiom,
    ! [W: int,Zb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Zb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Zb)) ).

% of_int_mult
tff(fact_4578_of__int__add,axiom,
    ! [W: int,Zb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Zb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Zb)) ).

% of_int_add
tff(fact_4579_of__int__add,axiom,
    ! [W: int,Zb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Zb)) = 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,Zb)) ).

% of_int_add
tff(fact_4580_of__int__add,axiom,
    ! [W: int,Zb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,W),Zb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Zb)) ).

% of_int_add
tff(fact_4581_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% real_sqrt_lt_0_iff
tff(fact_4582_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_4583_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_4584_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% real_sqrt_le_0_iff
tff(fact_4585_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_4586_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_4587_mask__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,zero_zero_nat) = zero_zero_nat ).

% mask_0
tff(fact_4588_mask__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,zero_zero_nat) = zero_zero_int ).

% mask_0
tff(fact_4589_of__int__diff,axiom,
    ! [W: int,Zb: int] : aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Zb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,W)),aa_int_real(ring_1_of_int_real,Zb)) ).

% of_int_diff
tff(fact_4590_of__int__diff,axiom,
    ! [W: int,Zb: int] : aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Zb)) = aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,W)),aa_int_rat(ring_1_of_int_rat,Zb)) ).

% of_int_diff
tff(fact_4591_of__int__diff,axiom,
    ! [W: int,Zb: int] : aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,W),Zb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(ring_1_of_int_int,W)),aa_int_int(ring_1_of_int_int,Zb)) ).

% of_int_diff
tff(fact_4592_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% real_sqrt_lt_1_iff
tff(fact_4593_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_4594_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% real_sqrt_le_1_iff
tff(fact_4595_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_4596_of__int__power,axiom,
    ! [Zb: int,Nb: nat] : aa_int_rat(ring_1_of_int_rat,aa_nat_int(power_power_int(Zb),Nb)) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,Zb)),Nb) ).

% of_int_power
tff(fact_4597_of__int__power,axiom,
    ! [Zb: int,Nb: nat] : aa_int_real(ring_1_of_int_real,aa_nat_int(power_power_int(Zb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,Zb)),Nb) ).

% of_int_power
tff(fact_4598_of__int__power,axiom,
    ! [Zb: int,Nb: nat] : aa_int_int(ring_1_of_int_int,aa_nat_int(power_power_int(Zb),Nb)) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,Zb)),Nb) ).

% of_int_power
tff(fact_4599_of__int__power,axiom,
    ! [Zb: int,Nb: nat] : aa_int_complex(ring_17405671764205052669omplex,aa_nat_int(power_power_int(Zb),Nb)) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,Zb)),Nb) ).

% of_int_power
tff(fact_4600_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) = aa_int_rat(ring_1_of_int_rat,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4601_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) = aa_int_real(ring_1_of_int_real,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4602_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) = aa_int_int(ring_1_of_int_int,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4603_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( ( aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B2)),W) = aa_int_complex(ring_17405671764205052669omplex,Xb) )
    <=> ( aa_nat_int(power_power_int(B2),W) = Xb ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_4604_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Xb) = aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4605_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4606_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Xb) = aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4607_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Xb) = aa_nat_complex(power_power_complex(aa_int_complex(ring_17405671764205052669omplex,B2)),W) )
    <=> ( Xb = aa_nat_int(power_power_int(B2),W) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_4608_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_4609_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_4610_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_4611_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_4612_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_4613_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_4614_and__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_4615_and__numerals_I8_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_4616_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_4617_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_4618_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_4619_take__bit__minus__one__eq__mask,axiom,
    ! [Nb: nat] : bit_se1745604003318907178nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_se2119862282449309892nteger(Nb) ).

% take_bit_minus_one_eq_mask
tff(fact_4620_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_4621_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% take_bit_of_Suc_0
tff(fact_4622_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_4623_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_4624_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_4625_and__numerals_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_4626_and__numerals_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(3)
tff(fact_4627_and__numerals_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(3)
tff(fact_4628_of__int__le__0__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Zb)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4629_of__int__le__0__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Zb)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4630_of__int__le__0__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Zb)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),zero_zero_int) ) ).

% of_int_le_0_iff
tff(fact_4631_of__int__0__le__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb) ) ).

% of_int_0_le_iff
tff(fact_4632_of__int__0__le__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb) ) ).

% of_int_0_le_iff
tff(fact_4633_of__int__0__le__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb) ) ).

% of_int_0_le_iff
tff(fact_4634_of__int__0__less__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb) ) ).

% of_int_0_less_iff
tff(fact_4635_of__int__0__less__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb) ) ).

% of_int_0_less_iff
tff(fact_4636_of__int__0__less__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb) ) ).

% of_int_0_less_iff
tff(fact_4637_of__int__less__0__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Zb)),zero_zero_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4638_of__int__less__0__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Zb)),zero_zero_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4639_of__int__less__0__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Zb)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),zero_zero_int) ) ).

% of_int_less_0_iff
tff(fact_4640_of__int__le__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Zb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4641_of__int__le__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Zb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4642_of__int__le__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Zb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_le_numeral_iff
tff(fact_4643_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_le_iff
tff(fact_4644_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_le_iff
tff(fact_4645_of__int__numeral__le__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_le_iff
tff(fact_4646_of__int__less__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Zb)),aa_num_real(numeral_numeral_real,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4647_of__int__less__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Zb)),aa_num_rat(numeral_numeral_rat,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4648_of__int__less__numeral__iff,axiom,
    ! [Zb: int,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Zb)),aa_num_int(numeral_numeral_int,Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_num_int(numeral_numeral_int,Nb)) ) ).

% of_int_less_numeral_iff
tff(fact_4649_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_less_iff
tff(fact_4650_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_less_iff
tff(fact_4651_of__int__numeral__less__iff,axiom,
    ! [Nb: num,Zb: 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,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,Nb)),Zb) ) ).

% of_int_numeral_less_iff
tff(fact_4652_of__int__le__1__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4653_of__int__le__1__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4654_of__int__le__1__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Zb)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),one_one_int) ) ).

% of_int_le_1_iff
tff(fact_4655_of__int__1__le__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(ring_1_of_int_real,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Zb) ) ).

% of_int_1_le_iff
tff(fact_4656_of__int__1__le__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Zb) ) ).

% of_int_1_le_iff
tff(fact_4657_of__int__1__le__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(ring_1_of_int_int,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Zb) ) ).

% of_int_1_le_iff
tff(fact_4658_of__int__less__1__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4659_of__int__less__1__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4660_of__int__less__1__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Zb)),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),one_one_int) ) ).

% of_int_less_1_iff
tff(fact_4661_of__int__1__less__iff,axiom,
    ! [Zb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_int_real(ring_1_of_int_real,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Zb) ) ).

% of_int_1_less_iff
tff(fact_4662_of__int__1__less__iff,axiom,
    ! [Zb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(ring_1_of_int_rat,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Zb) ) ).

% of_int_1_less_iff
tff(fact_4663_of__int__1__less__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(ring_1_of_int_int,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Zb) ) ).

% of_int_1_less_iff
tff(fact_4664_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_4665_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_4666_take__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),one_one_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_4667_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4668_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4669_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4670_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_4671_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4672_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4673_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4674_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_4675_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4676_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4677_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_4678_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4679_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4680_of__int__power__le__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_4681_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4682_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4683_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B2: int,W: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B2),W)),Xb) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_4684_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_int_real(ring_1_of_int_real,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4685_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Xb)),aa_nat_rat(power_power_rat(aa_int_rat(ring_1_of_int_rat,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4686_of__int__power__less__of__int__cancel__iff,axiom,
    ! [Xb: int,B2: int,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,Xb)),aa_nat_int(power_power_int(aa_int_int(ring_1_of_int_int,B2)),W))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(B2),W)) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_4687_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_4688_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_4689_and__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(4)
tff(fact_4690_and__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(4)
tff(fact_4691_and__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).

% and_numerals(6)
tff(fact_4692_and__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).

% and_numerals(6)
tff(fact_4693_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_4694_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_4695_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_4696_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_4697_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_4698_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_4699_take__bit__Suc__0,axiom,
    ! [A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,zero_zero_nat)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_0
tff(fact_4700_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_4701_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_4702_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_4703_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4704_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4705_numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_4706_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4707_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4708_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_4709_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4710_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4711_numeral__power__less__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_4712_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4713_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4714_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_4715_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_4716_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_real(ring_1_of_int_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4717_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_int(ring_1_of_int_int,Y) = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4718_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_complex(ring_17405671764205052669omplex,Y) = aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4719_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( ring_18347121197199848620nteger(Y) = power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4720_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_rat(ring_1_of_int_rat,Y) = aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_4721_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb) = aa_int_real(ring_1_of_int_real,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4722_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = aa_int_int(ring_1_of_int_int,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4723_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Xb))),Nb) = aa_int_complex(ring_17405671764205052669omplex,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4724_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb) = ring_18347121197199848620nteger(Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4725_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb) = aa_int_rat(ring_1_of_int_rat,Y) )
    <=> ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb) = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_4726_and__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).

% and_numerals(7)
tff(fact_4727_and__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% and_numerals(7)
tff(fact_4728_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : bit_se1745604003318907178nteger(Mb,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),Mb))),power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% take_bit_of_exp
tff(fact_4729_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_4730_take__bit__of__exp,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_of_exp
tff(fact_4731_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_4732_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(aa_int_fun_int_int(times_times_int,aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_of_2
tff(fact_4733_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,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_4734_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb)),aa_int_real(ring_1_of_int_real,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4735_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb)),ring_18347121197199848620nteger(A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4736_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb)),aa_int_rat(ring_1_of_int_rat,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4737_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),aa_int_int(ring_1_of_int_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)),A) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_4738_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4739_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,ring_18347121197199848620nteger(A)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4740_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4741_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,A)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Xb))),Nb)) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_4742_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_4743_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_4744_and_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(B2),aa_int_int(bit_se725231765392027082nd_int(A),C)) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B2),C)) ).

% and.left_commute
tff(fact_4745_and_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B2),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,B2),C)) ).

% and.left_commute
tff(fact_4746_and_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_se725231765392027082nd_int(A),B2) = aa_int_int(bit_se725231765392027082nd_int(B2),A) ).

% and.commute
tff(fact_4747_and_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B2),A) ).

% and.commute
tff(fact_4748_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_4749_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_4750_and_Oassoc,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se725231765392027082nd_int(A),B2)),C) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se725231765392027082nd_int(B2),C)) ).

% and.assoc
tff(fact_4751_and_Oassoc,axiom,
    ! [A: nat,B2: 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),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,B2),C)) ).

% and.assoc
tff(fact_4752_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_4753_take__bit__add,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B2))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) ).

% take_bit_add
tff(fact_4754_take__bit__add,axiom,
    ! [Nb: nat,A: nat,B2: 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),B2))) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) ).

% take_bit_add
tff(fact_4755_take__bit__tightened,axiom,
    ! [Nb: nat,A: int,B2: int,Mb: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_int(bit_se2923211474154528505it_int(Mb),A) = aa_int_int(bit_se2923211474154528505it_int(Mb),B2) ) ) ) ).

% take_bit_tightened
tff(fact_4756_take__bit__tightened,axiom,
    ! [Nb: nat,A: nat,B2: nat,Mb: 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),B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),B2) ) ) ) ).

% take_bit_tightened
tff(fact_4757_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb) ).

% take_bit_nat_less_eq_self
tff(fact_4758_take__bit__tightened__less__eq__nat,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4759_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_4760_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_4761_mult__of__int__commute,axiom,
    ! [Xb: int,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_int_real(ring_1_of_int_real,Xb)) ).

% mult_of_int_commute
tff(fact_4762_mult__of__int__commute,axiom,
    ! [Xb: int,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,Xb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_int_rat(ring_1_of_int_rat,Xb)) ).

% mult_of_int_commute
tff(fact_4763_mult__of__int__commute,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(ring_1_of_int_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_int_int(ring_1_of_int_int,Xb)) ).

% mult_of_int_commute
tff(fact_4764_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_divide
tff(fact_4765_real__sqrt__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y)) ).

% real_sqrt_mult
tff(fact_4766_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_4767_real__sqrt__minus,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_4768_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_4769_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,K),L)) ).

% take_bit_mult
tff(fact_4770_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),L))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(minus_minus_int,K),L)) ).

% take_bit_diff
tff(fact_4771_concat__bit__take__bit__eq,axiom,
    ! [Nb: nat,B2: int] : bit_concat_bit(Nb,aa_int_int(bit_se2923211474154528505it_int(Nb),B2)) = bit_concat_bit(Nb,B2) ).

% concat_bit_take_bit_eq
tff(fact_4772_concat__bit__eq__iff,axiom,
    ! [Nb: nat,K: int,L: int,R2: int,S: int] :
      ( ( aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(bit_concat_bit(Nb,R2),S) )
    <=> ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(bit_se2923211474154528505it_int(Nb),R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_4773_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_4774_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(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
tff(fact_4775_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( bit_se3949692690581998587nteger(A,B2) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_4776_and__eq__minus__1__iff,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),B2) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_4777_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_4778_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_4779_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sqrt,Xb) = zero_zero_real )
       => ( Xb = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_4780_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_4781_take__bit__tightened__less__eq__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se2923211474154528505it_int(Mb),K)),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4782_AND__upper2_H,axiom,
    ! [Y: int,Zb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),Zb) ) ) ).

% AND_upper2'
tff(fact_4783_AND__upper1_H,axiom,
    ! [Y: int,Zb: 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),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Zb) ) ) ).

% AND_upper1'
tff(fact_4784_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_4785_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_4786_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se725231765392027082nd_int(Xb),Y)) ) ).

% AND_lower
tff(fact_4787_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_4788_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_4789_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [Nb: nat,A: int,B2: int] :
      ( ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_ri631733984087533419it_int(Nb),B2) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),B2) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_4790_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_4791_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_4792_signed__take__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Mb),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),Mb),bit_se2923211474154528505it_int(Nb),bit_ri631733984087533419it_int(Mb)),A) ).

% signed_take_bit_take_bit
tff(fact_4793_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_4794_take__bit__unset__bit__eq,axiom,
    ! [Nb: nat,Mb: 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,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_unset_bit_eq
tff(fact_4795_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_4796_take__bit__set__bit__eq,axiom,
    ! [Nb: nat,Mb: 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,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_set_bit_eq
tff(fact_4797_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se2159334234014336723it_int(Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A),aa_int_int(bit_se2159334234014336723it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_4798_take__bit__flip__bit__eq,axiom,
    ! [Nb: nat,Mb: 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,Mb),A)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A))) ).

% take_bit_flip_bit_eq
tff(fact_4799_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_4800_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_4801_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,Xb)) = aa_real_real(sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_4802_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sqrt,Xb)),aa_real_real(sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_4803_take__bit__signed__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri631733984087533419it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(Mb),A) ) ) ).

% take_bit_signed_take_bit
tff(fact_4804_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_4805_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_4806_AND__upper1_H_H,axiom,
    ! [Y: int,Zb: 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),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Y),Ya)),Zb) ) ) ).

% AND_upper1''
tff(fact_4807_AND__upper2_H_H,axiom,
    ! [Y: int,Zb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),Zb) ) ) ).

% AND_upper2''
tff(fact_4808_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(aa_int_fun_int_int(minus_minus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)),one_one_int) ) ) ).

% take_bit_decr_eq
tff(fact_4809_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))) ).

% real_of_int_div4
tff(fact_4810_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(aa_int_fun_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_4811_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_4812_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(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_4813_even__and__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3949692690581998587nteger(A,B2))
    <=> ( 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))),B2) ) ) ).

% even_and_iff
tff(fact_4814_even__and__iff,axiom,
    ! [A: int,B2: 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),B2))
    <=> ( 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))),B2) ) ) ).

% even_and_iff
tff(fact_4815_even__and__iff,axiom,
    ! [A: nat,B2: 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),B2))
    <=> ( 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))),B2) ) ) ).

% even_and_iff
tff(fact_4816_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_4817_of__int__nonneg,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Zb)) ) ).

% of_int_nonneg
tff(fact_4818_of__int__nonneg,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Zb)) ) ).

% of_int_nonneg
tff(fact_4819_of__int__nonneg,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Zb)) ) ).

% of_int_nonneg
tff(fact_4820_of__int__pos,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_int_real(ring_1_of_int_real,Zb)) ) ).

% of_int_pos
tff(fact_4821_of__int__pos,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(ring_1_of_int_rat,Zb)) ) ).

% of_int_pos
tff(fact_4822_of__int__pos,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(ring_1_of_int_int,Zb)) ) ).

% of_int_pos
tff(fact_4823_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_4824_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_4825_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_4826_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_4827_of__int__neg__numeral,axiom,
    ! [K: num] : 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_4828_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_4829_int__le__real__less,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Mb)),one_one_real)) ) ).

% int_le_real_less
tff(fact_4830_int__less__real__le,axiom,
    ! [Nb: int,Mb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real)),aa_int_real(ring_1_of_int_real,Mb)) ) ).

% int_less_real_le
tff(fact_4831_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Xb)),aa_int_real(ring_1_of_int_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),D))),aa_int_real(ring_1_of_int_real,D))) ).

% real_of_int_div_aux
tff(fact_4832_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(aa_int_fun_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_4833_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_4834_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: code_integer] : bit_se1745604003318907178nteger(Nb,A) = modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) ).

% take_bit_eq_mod
tff(fact_4835_take__bit__eq__mod,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% take_bit_eq_mod
tff(fact_4836_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_4837_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_4838_one__and__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% one_and_eq
tff(fact_4839_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_4840_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_4841_and__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se725231765392027082nd_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% and_one_eq
tff(fact_4842_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_4843_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_4844_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_4845_take__bit__nat__eq__self,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = Mb ) ) ).

% take_bit_nat_eq_self
tff(fact_4846_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_4847_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_4848_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% sqrt_le_D
tff(fact_4849_take__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% take_bit_nat_def
tff(fact_4850_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_4851_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),K) = aa_int_int(aa_int_fun_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_4852_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))) ).

% real_of_int_div2
tff(fact_4853_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,Nb),Xb)))),one_one_real) ).

% real_of_int_div3
tff(fact_4854_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,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)),A) ) ).

% take_bit_eq_0_iff
tff(fact_4855_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_4856_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_4857_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(aa_int_fun_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_4858_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_4859_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)),Mb)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb) ) ).

% take_bit_nat_less_self_iff
tff(fact_4860_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Xb )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_4861_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_4862_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_4863_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_4864_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Y )
     => ( Xb = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_4865_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = Xb )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_4866_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_4867_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B2: 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,B2),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,B2),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_4868_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_4869_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_4870_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(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_Suc_minus_bit0
tff(fact_4871_even__of__int__iff,axiom,
    ! [K: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),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_4872_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_4873_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_4874_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_4875_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))),bit_se2119862282449309892nteger(Nb))
    <=> ( Nb = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_4876_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_4877_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_4878_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_4879_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_4880_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_4881_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_4882_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_4883_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% and_int_rec
tff(fact_4884_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_4885_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(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% take_bit_numeral_minus_bit0
tff(fact_4886_mask__half__int,axiom,
    ! [Nb: nat] : aa_int_int(aa_int_fun_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_4887_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),K) != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) )
     => ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_4888_mask__int__def,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_int_def
tff(fact_4889_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(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_4890_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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_4891_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_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_4892_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_4893_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(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_4894_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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_4895_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_4896_take__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% take_bit_Suc
tff(fact_4897_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_4898_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_4899_mask__eq__exp__minus__1,axiom,
    ! [Nb: nat] : aa_nat_int(bit_se2000444600071755411sk_int,Nb) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_4900_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sqrt,Xb)),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_4901_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xaa),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Ya),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_4902_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_4903_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(aa_int_fun_int_int(minus_minus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_4904_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(aa_int_fun_int_int(plus_plus_int,K),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_4905_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,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_4906_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(power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4907_stable__imp__take__bit__eq,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_int_int(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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4908_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_4909_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_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_4910_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_4911_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(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4912_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% one_le_exp_iff
tff(fact_4913_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% exp_le_one_iff
tff(fact_4914_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% exp_less_one_iff
tff(fact_4915_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_less_exp_iff
tff(fact_4916_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_4917_exp__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% exp_bound
tff(fact_4918_arsinh__real__aux,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_aux
tff(fact_4919_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_4920_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ).

% exp_less_mono
tff(fact_4921_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_4922_exp__zero,axiom,
    aa_complex_complex(exp_complex,zero_zero_complex) = one_one_complex ).

% exp_zero
tff(fact_4923_exp__zero,axiom,
    aa_real_real(exp_real,zero_zero_real) = one_one_real ).

% exp_zero
tff(fact_4924_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,Xb) = one_one_real )
    <=> ( Xb = zero_zero_real ) ) ).

% exp_eq_one_iff
tff(fact_4925_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = zero_zero_nat ).

% and_nat_numerals(3)
tff(fact_4926_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_4927_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_4928_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_4929_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_4930_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_4931_exp__times__arg__commute,axiom,
    ! [A3: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,A3)),A3) = aa_complex_complex(times_times_complex(A3),aa_complex_complex(exp_complex,A3)) ).

% exp_times_arg_commute
tff(fact_4932_exp__times__arg__commute,axiom,
    ! [A3: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,A3)),A3) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A3),aa_real_real(exp_real,A3)) ).

% exp_times_arg_commute
tff(fact_4933_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% exp_less_cancel
tff(fact_4934_and__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = $ite(
        ( ( Mb = zero_zero_nat )
        | ( Nb = zero_zero_nat ) ),
        zero_zero_nat,
        aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ).

% and_nat_unfold
tff(fact_4935_and__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% and_nat_rec
tff(fact_4936_mult__exp__exp,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) = aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% mult_exp_exp
tff(fact_4937_mult__exp__exp,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) = aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% mult_exp_exp
tff(fact_4938_exp__add__commuting,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(times_times_complex(Xb),Y) = aa_complex_complex(times_times_complex(Y),Xb) )
     => ( aa_complex_complex(exp_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ) ) ).

% exp_add_commuting
tff(fact_4939_exp__add__commuting,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) )
     => ( aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ) ) ).

% exp_add_commuting
tff(fact_4940_exp__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(exp_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,Y)) ).

% exp_diff
tff(fact_4941_exp__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,Y)) ).

% exp_diff
tff(fact_4942_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_less_zero
tff(fact_4943_exp__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_gt_zero
tff(fact_4944_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_4945_exp__ge__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(exp_real,Xb)) ).

% exp_ge_zero
tff(fact_4946_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Xb)),zero_zero_real) ).

% not_exp_le_zero
tff(fact_4947_exp__minus__inverse,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) = one_one_real ).

% exp_minus_inverse
tff(fact_4948_exp__minus__inverse,axiom,
    ! [Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) = one_one_complex ).

% exp_minus_inverse
tff(fact_4949_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(exp_real,Xb)) ) ).

% exp_gt_one
tff(fact_4950_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ).

% exp_ge_add_one_self
tff(fact_4951_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(exp_real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_4952_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_4953_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_4954_exp__double,axiom,
    ! [Zb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Zb)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_4955_exp__double,axiom,
    ! [Zb: 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))),Zb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% exp_double
tff(fact_4956_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_4957_arcosh__1,axiom,
    aa_real_real(arcosh_real,one_one_real) = zero_zero_real ).

% arcosh_1
tff(fact_4958_floor__exists1,axiom,
    ! [Xb: real] :
    ? [X3: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,X3)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,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)),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_4959_floor__exists1,axiom,
    ! [Xb: rat] :
    ? [X3: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,X3)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,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)),Xb)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Y4),one_one_int))) )
         => ( Y4 = X3 ) ) ) ).

% floor_exists1
tff(fact_4960_floor__exists,axiom,
    ! [Xb: real] :
    ? [Z: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Z)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))) ) ).

% floor_exists
tff(fact_4961_floor__exists,axiom,
    ! [Xb: rat] :
    ? [Z: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),one_one_int))) ) ).

% floor_exists
tff(fact_4962_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,K))))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_4963_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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_4964_tanh__real__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)))) ).

% tanh_real_altdef
tff(fact_4965_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_4966_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_4967_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_4968_ln__one,axiom,
    aa_real_real(ln_ln_real,one_one_real) = zero_zero_real ).

% ln_one
tff(fact_4969_ln__inj__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(ln_ln_real,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_4970_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_4971_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% tanh_real_neg_iff
tff(fact_4972_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% tanh_real_pos_iff
tff(fact_4973_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tanh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% tanh_real_nonpos_iff
tff(fact_4974_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tanh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_4975_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_4976_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_4977_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = zero_zero_real )
      <=> ( Xb = one_one_real ) ) ) ).

% ln_eq_zero_iff
tff(fact_4978_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_4979_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% ln_less_zero_iff
tff(fact_4980_exp__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb ) ) ).

% exp_ln
tff(fact_4981_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(exp_real,aa_real_real(ln_ln_real,Xb)) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% exp_ln_iff
tff(fact_4982_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% ln_le_zero_iff
tff(fact_4983_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_4984_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_4985_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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_4986_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_4987_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_4988_add__neg__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),aa_rat_rat(uminus_uminus_rat,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_4989_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,Mb))),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,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4990_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4991_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(Mb))),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4992_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Mb)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4993_add__neg__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,Mb))),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,Mb))) ).

% add_neg_numeral_special(6)
tff(fact_4994_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_4995_diff__numeral__special_I5_J,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) ).

% diff_numeral_special(5)
tff(fact_4996_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_4997_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_4998_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_4999_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,Mb)),aa_real_real(uminus_uminus_real,one_one_real)) = aa_num_real(numeral_numeral_real,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_5000_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_5001_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_complex_complex(minus_minus_complex(numera6690914467698888265omplex(Mb)),aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = numera6690914467698888265omplex(aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_5002_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(Mb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_5003_diff__numeral__special_I6_J,axiom,
    ! [Mb: num] : aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,Mb)),aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Mb)) ).

% diff_numeral_special(6)
tff(fact_5004_num__induct,axiom,
    ! [P: fun_num_o,Xb: 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,Xb) ) ) ).

% num_induct
tff(fact_5005_add__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(plus_plus_num(Xb),aa_num_num(inc,Y)) = aa_num_num(inc,aa_num_num(plus_plus_num(Xb),Y)) ).

% add_inc
tff(fact_5006_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_less_self
tff(fact_5007_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tanh_real,Xb)),one_one_real) ).

% tanh_real_lt_1
tff(fact_5008_inc_Osimps_I1_J,axiom,
    aa_num_num(inc,one) = aa_num_num(bit0,one) ).

% inc.simps(1)
tff(fact_5009_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit0,Xb)) = aa_num_num(bit1,Xb) ).

% inc.simps(2)
tff(fact_5010_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : aa_num_num(inc,aa_num_num(bit1,Xb)) = aa_num_num(bit0,aa_num_num(inc,Xb)) ).

% inc.simps(3)
tff(fact_5011_ln__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),Xb) ) ).

% ln_bound
tff(fact_5012_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_gt_zero
tff(fact_5013_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(ln_ln_real,Xb)),zero_zero_real) ) ) ).

% ln_less_zero
tff(fact_5014_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_5015_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb)) ) ).

% ln_ge_zero
tff(fact_5016_add__One,axiom,
    ! [Xb: num] : aa_num_num(plus_plus_num(Xb),one) = aa_num_num(inc,Xb) ).

% add_One
tff(fact_5017_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa_num_num(times_times_num(Xb),aa_num_num(inc,Y)) = aa_num_num(plus_plus_num(aa_num_num(times_times_num(Xb),Y)),Xb) ).

% mult_inc
tff(fact_5018_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(tanh_real,Xb)) ).

% tanh_real_gt_neg1
tff(fact_5019_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(ln_ln_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_5020_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_5021_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_mult
tff(fact_5022_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( ( aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real) )
       => ( Xb = one_one_real ) ) ) ).

% ln_eq_minus_one
tff(fact_5023_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y)) ) ) ) ).

% ln_div
tff(fact_5024_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(ln_ln_real,Xb))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_5025_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(exp_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Y)),Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_5026_numeral__inc,axiom,
    ! [Xb: num] : numera6690914467698888265omplex(aa_num_num(inc,Xb)) = aa_complex_complex(plus_plus_complex(numera6690914467698888265omplex(Xb)),one_one_complex) ).

% numeral_inc
tff(fact_5027_numeral__inc,axiom,
    ! [Xb: num] : aa_num_real(numeral_numeral_real,aa_num_num(inc,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,Xb)),one_one_real) ).

% numeral_inc
tff(fact_5028_numeral__inc,axiom,
    ! [Xb: num] : aa_num_rat(numeral_numeral_rat,aa_num_num(inc,Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,Xb)),one_one_rat) ).

% numeral_inc
tff(fact_5029_numeral__inc,axiom,
    ! [Xb: num] : aa_num_nat(numeral_numeral_nat,aa_num_num(inc,Xb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_num_nat(numeral_numeral_nat,Xb)),one_one_nat) ).

% numeral_inc
tff(fact_5030_numeral__inc,axiom,
    ! [Xb: num] : aa_num_int(numeral_numeral_int,aa_num_num(inc,Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Xb)),one_one_int) ).

% numeral_inc
tff(fact_5031_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_5032_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) ) ).

% ln_le_minus_one
tff(fact_5033_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_5034_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_5035_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(tanh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)) ) ) ).

% tanh_ln_real
tff(fact_5036_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ) ) ).

% arcosh_real_def
tff(fact_5037_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb))),aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_5038_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(sqrt,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% ln_sqrt
tff(fact_5039_exists__least__lemma,axiom,
    ! [P: fun_nat_o] :
      ( ~ aa_nat_o(P,zero_zero_nat)
     => ( ? [X_12: nat] : aa_nat_o(P,X_12)
       => ? [N2: nat] :
            ( ~ aa_nat_o(P,N2)
            & aa_nat_o(P,aa_nat_nat(suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_5040_ex__le__of__int,axiom,
    ! [Xb: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ).

% ex_le_of_int
tff(fact_5041_ex__le__of__int,axiom,
    ! [Xb: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ).

% ex_le_of_int
tff(fact_5042_ex__of__int__less,axiom,
    ! [Xb: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Z)),Xb) ).

% ex_of_int_less
tff(fact_5043_ex__of__int__less,axiom,
    ! [Xb: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Z)),Xb) ).

% ex_of_int_less
tff(fact_5044_ex__less__of__int,axiom,
    ! [Xb: real] :
    ? [Z: int] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Z)) ).

% ex_less_of_int
tff(fact_5045_ex__less__of__int,axiom,
    ! [Xb: rat] :
    ? [Z: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Z)) ).

% ex_less_of_int
tff(fact_5046_tanh__altdef,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)))) ).

% tanh_altdef
tff(fact_5047_tanh__altdef,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))),aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Xb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)))) ).

% tanh_altdef
tff(fact_5048_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_5049_artanh__def,axiom,
    ! [Xb: real] : aa_real_real(artanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% artanh_def
tff(fact_5050_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_5051_round__unique,axiom,
    ! [Xb: real,Y: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,Y))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => ( aa_real_int(archim8280529875227126926d_real,Xb) = Y ) ) ) ).

% round_unique
tff(fact_5052_round__unique,axiom,
    ! [Xb: rat,Y: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,Y))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))))
       => ( aa_rat_int(archim7778729529865785530nd_rat,Xb) = Y ) ) ) ).

% round_unique
tff(fact_5053_arsinh__real__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)))) ).

% arsinh_real_def
tff(fact_5054_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_5055_of__int__round__gt,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_gt
tff(fact_5056_of__int__round__gt,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_gt
tff(fact_5057_of__int__round__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))) ).

% of_int_round_ge
tff(fact_5058_of__int__round__ge,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))) ).

% of_int_round_ge
tff(fact_5059_abs__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(abs_abs_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_abs
tff(fact_5060_abs__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_abs
tff(fact_5061_abs__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_abs
tff(fact_5062_abs__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_abs
tff(fact_5063_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_5064_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_5065_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_5066_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_5067_abs__0,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_0
tff(fact_5068_abs__0,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_0
tff(fact_5069_abs__0,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_0
tff(fact_5070_abs__0,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_0
tff(fact_5071_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_5072_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_5073_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_5074_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_5075_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_5076_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_5077_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_5078_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_5079_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_5080_abs__zero,axiom,
    aa_real_real(abs_abs_real,zero_zero_real) = zero_zero_real ).

% abs_zero
tff(fact_5081_abs__zero,axiom,
    aa_rat_rat(abs_abs_rat,zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_5082_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_5083_abs__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(numera6620942414471956472nteger(Nb)) = numera6620942414471956472nteger(Nb) ).

% abs_numeral
tff(fact_5084_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_5085_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_5086_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_5087_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_5088_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_5089_abs__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,A)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) ).

% abs_mult_self_eq
tff(fact_5090_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_5091_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_5092_abs__1,axiom,
    aa_complex_complex(abs_abs_complex,one_one_complex) = one_one_complex ).

% abs_1
tff(fact_5093_abs__1,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_1
tff(fact_5094_abs__1,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_1
tff(fact_5095_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_5096_abs__add__abs,axiom,
    ! [A: code_integer,B2: code_integer] : abs_abs_Code_integer(plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2)) ).

% abs_add_abs
tff(fact_5097_abs__add__abs,axiom,
    ! [A: real,B2: 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,B2))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_add_abs
tff(fact_5098_abs__add__abs,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_add_abs
tff(fact_5099_abs__add__abs,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ).

% abs_add_abs
tff(fact_5100_abs__divide,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(abs_abs_complex,B2)) ).

% abs_divide
tff(fact_5101_abs__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_divide
tff(fact_5102_abs__divide,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_divide
tff(fact_5103_abs__minus,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(abs_abs_real,A) ).

% abs_minus
tff(fact_5104_abs__minus,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus
tff(fact_5105_abs__minus,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(abs_abs_complex,A) ).

% abs_minus
tff(fact_5106_abs__minus,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus
tff(fact_5107_abs__minus,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(abs_abs_rat,A) ).

% abs_minus
tff(fact_5108_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_5109_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_5110_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_5111_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_5112_abs__dvd__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(abs_abs_real,Mb)),K)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% abs_dvd_iff
tff(fact_5113_abs__dvd__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(abs_abs_int,Mb)),K)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% abs_dvd_iff
tff(fact_5114_abs__dvd__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,abs_abs_Code_integer(Mb)),K)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% abs_dvd_iff
tff(fact_5115_abs__dvd__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(abs_abs_rat,Mb)),K)
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% abs_dvd_iff
tff(fact_5116_dvd__abs__iff,axiom,
    ! [Mb: real,K: real] :
      ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),aa_real_real(abs_abs_real,K))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Mb),K) ) ).

% dvd_abs_iff
tff(fact_5117_dvd__abs__iff,axiom,
    ! [Mb: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),aa_int_int(abs_abs_int,K))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Mb),K) ) ).

% dvd_abs_iff
tff(fact_5118_dvd__abs__iff,axiom,
    ! [Mb: code_integer,K: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),abs_abs_Code_integer(K))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Mb),K) ) ).

% dvd_abs_iff
tff(fact_5119_dvd__abs__iff,axiom,
    ! [Mb: rat,K: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),aa_rat_rat(abs_abs_rat,K))
    <=> aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,Mb),K) ) ).

% dvd_abs_iff
tff(fact_5120_abs__bool__eq,axiom,
    ! [P: $o] : aa_real_real(abs_abs_real,zero_n3304061248610475627l_real((P))) = zero_n3304061248610475627l_real((P)) ).

% abs_bool_eq
tff(fact_5121_abs__bool__eq,axiom,
    ! [P: $o] : aa_rat_rat(abs_abs_rat,zero_n2052037380579107095ol_rat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% abs_bool_eq
tff(fact_5122_abs__bool__eq,axiom,
    ! [P: $o] : aa_int_int(abs_abs_int,aa_o_int(zero_n2684676970156552555ol_int,(P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% abs_bool_eq
tff(fact_5123_abs__bool__eq,axiom,
    ! [P: $o] : abs_abs_Code_integer(zero_n356916108424825756nteger((P))) = zero_n356916108424825756nteger((P)) ).

% abs_bool_eq
tff(fact_5124_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_5125_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_5126_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_5127_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_5128_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_5129_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_5130_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_5131_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_5132_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_5133_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_5134_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_5135_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_5136_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_5137_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_5138_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_5139_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_5140_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_5141_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_5142_abs__neg__numeral,axiom,
    ! [Nb: num] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(Nb) ).

% abs_neg_numeral
tff(fact_5143_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_5144_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_5145_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_5146_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_5147_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_5148_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_5149_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_5150_abs__power__minus,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),Nb)) = abs_abs_Code_integer(power_8256067586552552935nteger(A,Nb)) ).

% abs_power_minus
tff(fact_5151_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_5152_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Xb)) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs2
tff(fact_5153_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_5154_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_5155_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_5156_round__1,axiom,
    aa_real_int(archim8280529875227126926d_real,one_one_real) = one_one_int ).

% round_1
tff(fact_5157_round__1,axiom,
    aa_rat_int(archim7778729529865785530nd_rat,one_one_rat) = one_one_int ).

% round_1
tff(fact_5158_zero__le__divide__abs__iff,axiom,
    ! [A: real,B2: 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,B2)))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        | ( B2 = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5159_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B2: 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,B2)))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        | ( B2 = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_5160_divide__le__0__abs__iff,axiom,
    ! [A: real,B2: 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,B2))),zero_zero_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real)
        | ( B2 = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5161_divide__le__0__abs__iff,axiom,
    ! [A: rat,B2: 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,B2))),zero_zero_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat)
        | ( B2 = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
tff(fact_5162_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_5163_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_5164_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_5165_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_5166_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(artanh_real,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(artanh_real,Xb)) ) ) ).

% artanh_minus_real
tff(fact_5167_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A),Nb))
    <=> ( ( A != zero_z3403309356797280102nteger )
        | ( Nb = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
tff(fact_5168_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_5169_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_5170_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_5171_abs__power2,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(power_8256067586552552935nteger(A,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = power_8256067586552552935nteger(A,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% abs_power2
tff(fact_5172_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_5173_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_5174_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_5175_power2__abs,axiom,
    ! [A: code_integer] : power_8256067586552552935nteger(abs_abs_Code_integer(A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = power_8256067586552552935nteger(A,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% power2_abs
tff(fact_5176_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_5177_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_5178_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_5179_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_5180_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_5181_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))
     => ( power_8256067586552552935nteger(abs_abs_Code_integer(A),aa_num_nat(numeral_numeral_nat,W)) = power_8256067586552552935nteger(A,aa_num_nat(numeral_numeral_nat,W)) ) ) ).

% power_even_abs_numeral
tff(fact_5182_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_5183_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_5184_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_5185_real__sqrt__abs,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% real_sqrt_abs
tff(fact_5186_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_5187_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_5188_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_5189_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_5190_abs__le__D1,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% abs_le_D1
tff(fact_5191_abs__le__D1,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2) ) ).

% abs_le_D1
tff(fact_5192_abs__le__D1,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% abs_le_D1
tff(fact_5193_abs__le__D1,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% abs_le_D1
tff(fact_5194_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
tff(fact_5195_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_5196_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_5197_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_5198_abs__mult,axiom,
    ! [A: code_integer,B2: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B2)) ).

% abs_mult
tff(fact_5199_abs__mult,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ).

% abs_mult
tff(fact_5200_abs__mult,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ).

% abs_mult
tff(fact_5201_abs__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ).

% abs_mult
tff(fact_5202_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_5203_abs__one,axiom,
    aa_real_real(abs_abs_real,one_one_real) = one_one_real ).

% abs_one
tff(fact_5204_abs__one,axiom,
    aa_rat_rat(abs_abs_rat,one_one_rat) = one_one_rat ).

% abs_one
tff(fact_5205_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_5206_abs__minus__commute,axiom,
    ! [A: code_integer,B2: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A)) ).

% abs_minus_commute
tff(fact_5207_abs__minus__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) ).

% abs_minus_commute
tff(fact_5208_abs__minus__commute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B2)) = aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B2),A)) ).

% abs_minus_commute
tff(fact_5209_abs__minus__commute,axiom,
    ! [A: int,B2: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A)) ).

% abs_minus_commute
tff(fact_5210_abs__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(abs_abs_real,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_real_real(uminus_uminus_real,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5211_abs__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(abs_abs_int,Xb) = aa_int_int(abs_abs_int,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_int_int(uminus_uminus_int,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5212_abs__eq__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( ( abs_abs_Code_integer(Xb) = abs_abs_Code_integer(Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5213_abs__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( aa_rat_rat(abs_abs_rat,Xb) = aa_rat_rat(abs_abs_rat,Y) )
    <=> ( ( Xb = Y )
        | ( Xb = aa_rat_rat(uminus_uminus_rat,Y) ) ) ) ).

% abs_eq_iff
tff(fact_5214_power__abs,axiom,
    ! [A: code_integer,Nb: nat] : abs_abs_Code_integer(power_8256067586552552935nteger(A,Nb)) = power_8256067586552552935nteger(abs_abs_Code_integer(A),Nb) ).

% power_abs
tff(fact_5215_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_5216_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_5217_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_5218_dvd__if__abs__eq,axiom,
    ! [L: real,K: real] :
      ( ( aa_real_real(abs_abs_real,L) = aa_real_real(abs_abs_real,K) )
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5219_dvd__if__abs__eq,axiom,
    ! [L: int,K: int] :
      ( ( aa_int_int(abs_abs_int,L) = aa_int_int(abs_abs_int,K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5220_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( abs_abs_Code_integer(L) = abs_abs_Code_integer(K) )
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5221_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( aa_rat_rat(abs_abs_rat,L) = aa_rat_rat(abs_abs_rat,K) )
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,L),K) ) ).

% dvd_if_abs_eq
tff(fact_5222_round__diff__minimal,axiom,
    ! [Zb: real,Mb: int] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Zb),aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Zb))))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Zb),aa_int_real(ring_1_of_int_real,Mb)))) ).

% round_diff_minimal
tff(fact_5223_round__diff__minimal,axiom,
    ! [Zb: rat,Mb: int] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Zb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Zb))))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Zb),aa_int_rat(ring_1_of_int_rat,Mb)))) ).

% round_diff_minimal
tff(fact_5224_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_5225_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_5226_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_5227_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_5228_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_5229_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_5230_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_5231_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_5232_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_5233_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_5234_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_5235_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_5236_abs__triangle__ineq,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq
tff(fact_5237_abs__triangle__ineq,axiom,
    ! [A: real,B2: 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),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq
tff(fact_5238_abs__triangle__ineq,axiom,
    ! [A: rat,B2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq
tff(fact_5239_abs__triangle__ineq,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq
tff(fact_5240_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B2: 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(B2)),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(B2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D)) ) ) ).

% abs_mult_less
tff(fact_5241_abs__mult__less,axiom,
    ! [A: real,C: real,B2: 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,B2)),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,B2))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,C),D)) ) ) ).

% abs_mult_less
tff(fact_5242_abs__mult__less,axiom,
    ! [A: rat,C: rat,B2: 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,B2)),D)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D)) ) ) ).

% abs_mult_less
tff(fact_5243_abs__mult__less,axiom,
    ! [A: int,C: int,B2: 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,B2)),D)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D)) ) ) ).

% abs_mult_less
tff(fact_5244_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))) ).

% abs_triangle_ineq2
tff(fact_5245_abs__triangle__ineq2,axiom,
    ! [A: real,B2: 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,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% abs_triangle_ineq2
tff(fact_5246_abs__triangle__ineq2,axiom,
    ! [A: rat,B2: 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,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B2))) ).

% abs_triangle_ineq2
tff(fact_5247_abs__triangle__ineq2,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))) ).

% abs_triangle_ineq2
tff(fact_5248_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B2: 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(B2)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))) ).

% abs_triangle_ineq3
tff(fact_5249_abs__triangle__ineq3,axiom,
    ! [A: real,B2: 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,B2)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% abs_triangle_ineq3
tff(fact_5250_abs__triangle__ineq3,axiom,
    ! [A: rat,B2: 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,B2)))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(A),B2))) ).

% abs_triangle_ineq3
tff(fact_5251_abs__triangle__ineq3,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))) ).

% abs_triangle_ineq3
tff(fact_5252_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,A))) ).

% abs_triangle_ineq2_sym
tff(fact_5253_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B2: 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,B2))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5254_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B2: 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,B2))),aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5255_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),A))) ).

% abs_triangle_ineq2_sym
tff(fact_5256_nonzero__abs__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_5257_nonzero__abs__divide,axiom,
    ! [B2: rat,A: rat] :
      ( ( B2 != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% nonzero_abs_divide
tff(fact_5258_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_5259_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_5260_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_5261_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_5262_abs__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5263_abs__le__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5264_abs__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5265_abs__le__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2) ) ) ).

% abs_le_iff
tff(fact_5266_abs__le__D2,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2) ) ).

% abs_le_D2
tff(fact_5267_abs__le__D2,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ).

% abs_le_D2
tff(fact_5268_abs__le__D2,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ).

% abs_le_D2
tff(fact_5269_abs__le__D2,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2) ) ).

% abs_le_D2
tff(fact_5270_abs__leI,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,A)),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,A)),B2) ) ) ).

% abs_leI
tff(fact_5271_abs__leI,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,A),B2)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2)
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(A)),B2) ) ) ).

% abs_leI
tff(fact_5272_abs__leI,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,A)),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,A)),B2) ) ) ).

% abs_leI
tff(fact_5273_abs__leI,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B2) ) ) ).

% abs_leI
tff(fact_5274_abs__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,A)),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5275_abs__less__iff,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5276_abs__less__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(A)),B2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),B2)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5277_abs__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,A)),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,A)),B2) ) ) ).

% abs_less_iff
tff(fact_5278_dense__eq0__I,axiom,
    ! [Xb: real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),E2) )
     => ( Xb = zero_zero_real ) ) ).

% dense_eq0_I
tff(fact_5279_dense__eq0__I,axiom,
    ! [Xb: rat] :
      ( ! [E2: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E2)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),E2) )
     => ( Xb = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_5280_abs__eq__mult,axiom,
    ! [A: code_integer,B2: 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),B2)
          | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,B2),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B2)) ) ) ).

% abs_eq_mult
tff(fact_5281_abs__eq__mult,axiom,
    ! [A: real,B2: 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),B2)
          | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real) ) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% abs_eq_mult
tff(fact_5282_abs__eq__mult,axiom,
    ! [A: rat,B2: 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),B2)
          | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),zero_zero_rat) ) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% abs_eq_mult
tff(fact_5283_abs__eq__mult,axiom,
    ! [A: int,B2: 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),B2)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ) ) ).

% abs_eq_mult
tff(fact_5284_abs__mult__pos,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Xb)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),Xb) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5285_abs__mult__pos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,Y)),Xb) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5286_abs__mult__pos,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,Y)),Xb) = aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5287_abs__mult__pos,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),Xb) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),Xb)) ) ) ).

% abs_mult_pos
tff(fact_5288_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_5289_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_5290_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_5291_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_5292_eq__abs__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( A = aa_real_real(abs_abs_real,B2) )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
        & ( ( B2 = A )
          | ( B2 = aa_real_real(uminus_uminus_real,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5293_eq__abs__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( A = abs_abs_Code_integer(B2) )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),A)
        & ( ( B2 = A )
          | ( B2 = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5294_eq__abs__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = aa_rat_rat(abs_abs_rat,B2) )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
        & ( ( B2 = A )
          | ( B2 = aa_rat_rat(uminus_uminus_rat,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5295_eq__abs__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( A = aa_int_int(abs_abs_int,B2) )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
        & ( ( B2 = A )
          | ( B2 = aa_int_int(uminus_uminus_int,A) ) ) ) ) ).

% eq_abs_iff'
tff(fact_5296_abs__eq__iff_H,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(abs_abs_real,A) = B2 )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),B2)
        & ( ( A = B2 )
          | ( A = aa_real_real(uminus_uminus_real,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5297_abs__eq__iff_H,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ( abs_abs_Code_integer(A) = B2 )
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),B2)
        & ( ( A = B2 )
          | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5298_abs__eq__iff_H,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(abs_abs_rat,A) = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),B2)
        & ( ( A = B2 )
          | ( A = aa_rat_rat(uminus_uminus_rat,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5299_abs__eq__iff_H,axiom,
    ! [A: int,B2: int] :
      ( ( aa_int_int(abs_abs_int,A) = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
        & ( ( A = B2 )
          | ( A = aa_int_int(uminus_uminus_int,B2) ) ) ) ) ).

% abs_eq_iff'
tff(fact_5300_abs__div__pos,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(abs_abs_real,Xb)),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_5301_abs__div__pos,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Y)
     => ( aa_rat_rat(divide_divide_rat(aa_rat_rat(abs_abs_rat,Xb)),Y) = aa_rat_rat(abs_abs_rat,aa_rat_rat(divide_divide_rat(Xb),Y)) ) ) ).

% abs_div_pos
tff(fact_5302_zero__le__power__abs,axiom,
    ! [A: code_integer,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),power_8256067586552552935nteger(abs_abs_Code_integer(A),Nb)) ).

% zero_le_power_abs
tff(fact_5303_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_5304_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_5305_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_5306_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_5307_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_5308_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_5309_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_5310_abs__if__raw,axiom,
    ! [X4: real] :
      aa_real_real(abs_abs_real,X4) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,X4),zero_zero_real),aa_real_real(uminus_uminus_real,X4),X4) ).

% abs_if_raw
tff(fact_5311_abs__if__raw,axiom,
    ! [X4: int] :
      aa_int_int(abs_abs_int,X4) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,X4),zero_zero_int),aa_int_int(uminus_uminus_int,X4),X4) ).

% abs_if_raw
tff(fact_5312_abs__if__raw,axiom,
    ! [X4: code_integer] :
      abs_abs_Code_integer(X4) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,X4),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X4),X4) ).

% abs_if_raw
tff(fact_5313_abs__if__raw,axiom,
    ! [X4: rat] :
      aa_rat_rat(abs_abs_rat,X4) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X4),zero_zero_rat),aa_rat_rat(uminus_uminus_rat,X4),X4) ).

% abs_if_raw
tff(fact_5314_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_5315_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_5316_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_5317_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_5318_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,B2))),plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2))) ).

% abs_triangle_ineq4
tff(fact_5319_abs__triangle__ineq4,axiom,
    ! [A: real,B2: 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),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2))) ).

% abs_triangle_ineq4
tff(fact_5320_abs__triangle__ineq4,axiom,
    ! [A: rat,B2: 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),B2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2))) ).

% abs_triangle_ineq4
tff(fact_5321_abs__triangle__ineq4,axiom,
    ! [A: int,B2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))) ).

% abs_triangle_ineq4
tff(fact_5322_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B2: 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,B2),plus_p5714425477246183910nteger(C,D)))),plus_p5714425477246183910nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,C)),abs_abs_Code_integer(minus_8373710615458151222nteger(B2,D)))) ).

% abs_diff_triangle_ineq
tff(fact_5323_abs__diff__triangle__ineq,axiom,
    ! [A: real,B2: 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),B2)),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,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5324_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B2: 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(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D)))),aa_rat_rat(aa_rat_fun_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(B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5325_abs__diff__triangle__ineq,axiom,
    ! [A: int,B2: int,C: int,D: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),D)))) ).

% abs_diff_triangle_ineq
tff(fact_5326_abs__diff__le__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5327_abs__diff__le__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5328_abs__diff__le__iff,axiom,
    ! [Xb: 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(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5329_abs__diff__le__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_le_iff
tff(fact_5330_abs__diff__less__iff,axiom,
    ! [Xb: code_integer,A: code_integer,R2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,A))),R2)
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,minus_8373710615458151222nteger(A,R2)),Xb)
        & aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),plus_p5714425477246183910nteger(A,R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5331_abs__diff__less__iff,axiom,
    ! [Xb: real,A: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A))),R2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),R2)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5332_abs__diff__less__iff,axiom,
    ! [Xb: 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(Xb),A))),R2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(A),R2)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5333_abs__diff__less__iff,axiom,
    ! [Xb: int,A: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),A))),R2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),R2)),Xb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),R2)) ) ) ).

% abs_diff_less_iff
tff(fact_5334_round__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim7778729529865785530nd_rat,Xb)),aa_rat_int(archim7778729529865785530nd_rat,Y)) ) ).

% round_mono
tff(fact_5335_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_5336_lemma__interval__lt,axiom,
    ! [A: real,Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_5337_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,U)),V)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_5338_abs__add__one__gt__zero,axiom,
    ! [Xb: code_integer] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),plus_p5714425477246183910nteger(one_one_Code_integer,abs_abs_Code_integer(Xb))) ).

% abs_add_one_gt_zero
tff(fact_5339_abs__add__one__gt__zero,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(abs_abs_real,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5340_abs__add__one__gt__zero,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5341_abs__add__one__gt__zero,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(abs_abs_int,Xb))) ).

% abs_add_one_gt_zero
tff(fact_5342_of__int__leD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_leD
tff(fact_5343_of__int__leD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% of_int_leD
tff(fact_5344_of__int__leD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% of_int_leD
tff(fact_5345_of__int__leD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),Xb) ) ) ).

% of_int_leD
tff(fact_5346_of__int__lessD,axiom,
    ! [Nb: int,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(ring_18347121197199848620nteger(Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,one_one_Code_integer),Xb) ) ) ).

% of_int_lessD
tff(fact_5347_of__int__lessD,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_int_real(ring_1_of_int_real,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% of_int_lessD
tff(fact_5348_of__int__lessD,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_int_rat(ring_1_of_int_rat,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% of_int_lessD
tff(fact_5349_of__int__lessD,axiom,
    ! [Nb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(ring_1_of_int_int,Nb))),Xb)
     => ( ( Nb = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Xb) ) ) ).

% of_int_lessD
tff(fact_5350_lemma__interval,axiom,
    ! [A: real,Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [Y4: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y4))),D2)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_5351_of__int__round__abs__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_5352_of__int__round__abs__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),Xb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% of_int_round_abs_le
tff(fact_5353_round__unique_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Nb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_int(archim8280529875227126926d_real,Xb) = Nb ) ) ).

% round_unique'
tff(fact_5354_round__unique_H,axiom,
    ! [Xb: 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(Xb),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,Xb) = Nb ) ) ).

% round_unique'
tff(fact_5355_abs__le__square__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),abs_abs_Code_integer(Y))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5356_abs__le__square__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5357_abs__le__square__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),aa_rat_rat(abs_abs_rat,Y))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5358_abs__le__square__iff,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_le_square_iff
tff(fact_5359_abs__square__eq__1,axiom,
    ! [Xb: code_integer] :
      ( ( power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(Xb) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_5360_abs__square__eq__1,axiom,
    ! [Xb: rat] :
      ( ( aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_rat )
    <=> ( aa_rat_rat(abs_abs_rat,Xb) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_5361_abs__square__eq__1,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
    <=> ( aa_real_real(abs_abs_real,Xb) = one_one_real ) ) ).

% abs_square_eq_1
tff(fact_5362_abs__square__eq__1,axiom,
    ! [Xb: int] :
      ( ( aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_5363_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)
     => ( power_8256067586552552935nteger(abs_abs_Code_integer(A),Nb) = power_8256067586552552935nteger(A,Nb) ) ) ).

% power_even_abs
tff(fact_5364_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_5365_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_5366_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_5367_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,zero_z3403309356797280102nteger),Y)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),power_8256067586552552935nteger(Y,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5368_power2__le__iff__abs__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5369_power2__le__iff__abs__le,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_rat(power_power_rat(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5370_power2__le__iff__abs__le,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_int(power_power_int(Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),Y) ) ) ).

% power2_le_iff_abs_le
tff(fact_5371_abs__square__le__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_5372_abs__square__le__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_le_1
tff(fact_5373_abs__square__le__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_5374_abs__square__le__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_le_1
tff(fact_5375_abs__square__less__1,axiom,
    ! [Xb: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_5376_abs__square__less__1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real) ) ).

% abs_square_less_1
tff(fact_5377_abs__square__less__1,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_5378_abs__square__less__1,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int) ) ).

% abs_square_less_1
tff(fact_5379_power__mono__even,axiom,
    ! [Nb: nat,A: code_integer,B2: 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(B2))
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,power_8256067586552552935nteger(A,Nb)),power_8256067586552552935nteger(B2,Nb)) ) ) ).

% power_mono_even
tff(fact_5380_power__mono__even,axiom,
    ! [Nb: nat,A: real,B2: 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,B2))
       => 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,B2),Nb)) ) ) ).

% power_mono_even
tff(fact_5381_power__mono__even,axiom,
    ! [Nb: nat,A: rat,B2: 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,B2))
       => 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(B2),Nb)) ) ) ).

% power_mono_even
tff(fact_5382_power__mono__even,axiom,
    ! [Nb: nat,A: int,B2: 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,B2))
       => 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(B2),Nb)) ) ) ).

% power_mono_even
tff(fact_5383_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Y) ) ).

% sqrt_ge_absD
tff(fact_5384_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs1
tff(fact_5385_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% real_sqrt_ge_abs2
tff(fact_5386_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_5387_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),one_one_real) ).

% cos_x_y_le_one
tff(fact_5388_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,U),aa_real_real(sqrt,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_5389_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb))),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_5390_of__int__round__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim8280529875227126926d_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5391_of__int__round__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim7778729529865785530nd_rat,Xb))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% of_int_round_le
tff(fact_5392_abs__sqrt__wlog,axiom,
    ! [P: fun_Co2081214437782444048eger_o,Xb: 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),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(Xb)),power_8256067586552552935nteger(Xb,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5393_abs__sqrt__wlog,axiom,
    ! [P: fun_real_fun_real_o,Xb: 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,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5394_abs__sqrt__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,Xb: 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,Xb)),aa_nat_rat(power_power_rat(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5395_abs__sqrt__wlog,axiom,
    ! [P: fun_int_fun_int_o,Xb: 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,Xb)),aa_nat_int(power_power_int(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ).

% abs_sqrt_wlog
tff(fact_5396_arctan__double,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(arctan,Xb)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% arctan_double
tff(fact_5397_arctan__half,axiom,
    ! [Xb: real] : aa_real_real(arctan,Xb) = 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,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% arctan_half
tff(fact_5398_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq1
tff(fact_5399_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_ri631733984087533419it_int(Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),K)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_nat(suc,Nb))),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5400_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit1,aa_num_num(bit0,one))))),aa_real_real(exp_real,one_one_real))),aa_real_real(ln_ln_real,Xb)) ) ) ).

% log_base_10_eq2
tff(fact_5401_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_5402_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = aa_nat_complex(semiri8010041392384452111omplex,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5403_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(semiri5074537144036343181t_real,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5404_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = aa_nat_rat(semiri681578069525770553at_rat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5405_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = aa_nat_nat(semiri1316708129612266289at_nat,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5406_of__nat__eq__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_nat_int(semiri1314217659103216013at_int,Nb) )
    <=> ( Mb = Nb ) ) ).

% of_nat_eq_iff
tff(fact_5407_abs__of__nat,axiom,
    ! [Nb: nat] : abs_abs_Code_integer(semiri4939895301339042750nteger(Nb)) = semiri4939895301339042750nteger(Nb) ).

% abs_of_nat
tff(fact_5408_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_5409_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_5410_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_5411_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Mb) = zero_zero_complex )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5412_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = zero_zero_real )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5413_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Mb) = zero_zero_rat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5414_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Mb) = zero_zero_nat )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5415_of__nat__eq__0__iff,axiom,
    ! [Mb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = zero_zero_int )
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
tff(fact_5416_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_5417_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_5418_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_5419_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_5420_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_5421_of__nat__0,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,zero_zero_nat) = zero_zero_complex ).

% of_nat_0
tff(fact_5422_of__nat__0,axiom,
    aa_nat_real(semiri5074537144036343181t_real,zero_zero_nat) = zero_zero_real ).

% of_nat_0
tff(fact_5423_of__nat__0,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,zero_zero_nat) = zero_zero_rat ).

% of_nat_0
tff(fact_5424_of__nat__0,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,zero_zero_nat) = zero_zero_nat ).

% of_nat_0
tff(fact_5425_of__nat__0,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% of_nat_0
tff(fact_5426_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5427_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5428_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5429_of__nat__less__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_iff
tff(fact_5430_of__nat__numeral,axiom,
    ! [Nb: num] : aa_nat_complex(semiri8010041392384452111omplex,aa_num_nat(numeral_numeral_nat,Nb)) = numera6690914467698888265omplex(Nb) ).

% of_nat_numeral
tff(fact_5431_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_5432_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_5433_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_5434_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_5435_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5436_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5437_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5438_of__nat__le__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% of_nat_le_iff
tff(fact_5439_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_add
tff(fact_5440_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_add
tff(fact_5441_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_add
tff(fact_5442_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_add
tff(fact_5443_of__nat__add,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_add
tff(fact_5444_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% of_nat_mult
tff(fact_5445_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_mult
tff(fact_5446_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_mult
tff(fact_5447_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mult
tff(fact_5448_of__nat__mult,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mult
tff(fact_5449_of__nat__1,axiom,
    aa_nat_complex(semiri8010041392384452111omplex,one_one_nat) = one_one_complex ).

% of_nat_1
tff(fact_5450_of__nat__1,axiom,
    aa_nat_real(semiri5074537144036343181t_real,one_one_nat) = one_one_real ).

% of_nat_1
tff(fact_5451_of__nat__1,axiom,
    aa_nat_rat(semiri681578069525770553at_rat,one_one_nat) = one_one_rat ).

% of_nat_1
tff(fact_5452_of__nat__1,axiom,
    aa_nat_nat(semiri1316708129612266289at_nat,one_one_nat) = one_one_nat ).

% of_nat_1
tff(fact_5453_of__nat__1,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% of_nat_1
tff(fact_5454_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_5455_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_5456_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_5457_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_5458_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_5459_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_5460_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_5461_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_5462_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_5463_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_5464_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),Nb) ).

% of_nat_power
tff(fact_5465_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),Nb) ).

% of_nat_power
tff(fact_5466_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),Nb) ).

% of_nat_power
tff(fact_5467_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb) ).

% of_nat_power
tff(fact_5468_of__nat__power,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Mb),Nb)) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb) ).

% of_nat_power
tff(fact_5469_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B2)),W) = aa_nat_complex(semiri8010041392384452111omplex,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_5470_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) = aa_nat_real(semiri5074537144036343181t_real,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_5471_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W) = aa_nat_rat(semiri681578069525770553at_rat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_5472_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) = aa_nat_nat(semiri1316708129612266289at_nat,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_5473_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( ( aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) = aa_nat_int(semiri1314217659103216013at_int,Xb) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) = Xb ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_5474_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Xb) = aa_nat_complex(power_power_complex(aa_nat_complex(semiri8010041392384452111omplex,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_5475_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_5476_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Xb) = aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_5477_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Xb) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_5478_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Xb) = aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W) )
    <=> ( Xb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_5479_zabs__less__one__iff,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Zb)),one_one_int)
    <=> ( Zb = zero_zero_int ) ) ).

% zabs_less_one_iff
tff(fact_5480_log__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),one_one_real) = zero_zero_real ).

% log_one
tff(fact_5481_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% arctan_less_zero_iff
tff(fact_5482_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_arctan_iff
tff(fact_5483_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(arctan,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_arctan_iff
tff(fact_5484_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% arctan_le_zero_iff
tff(fact_5485_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_complex(semiri8010041392384452111omplex,zero_n2687167440665602831ol_nat((P))) = zero_n1201886186963655149omplex((P)) ).

% of_nat_of_bool
tff(fact_5486_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_real(semiri5074537144036343181t_real,zero_n2687167440665602831ol_nat((P))) = zero_n3304061248610475627l_real((P)) ).

% of_nat_of_bool
tff(fact_5487_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_rat(semiri681578069525770553at_rat,zero_n2687167440665602831ol_nat((P))) = zero_n2052037380579107095ol_rat((P)) ).

% of_nat_of_bool
tff(fact_5488_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_nat(semiri1316708129612266289at_nat,zero_n2687167440665602831ol_nat((P))) = zero_n2687167440665602831ol_nat((P)) ).

% of_nat_of_bool
tff(fact_5489_of__nat__of__bool,axiom,
    ! [P: $o] : aa_nat_int(semiri1314217659103216013at_int,zero_n2687167440665602831ol_nat((P))) = aa_o_int(zero_n2684676970156552555ol_int,(P)) ).

% of_nat_of_bool
tff(fact_5490_of__nat__of__bool,axiom,
    ! [P: $o] : semiri4939895301339042750nteger(zero_n2687167440665602831ol_nat((P))) = zero_n356916108424825756nteger((P)) ).

% of_nat_of_bool
tff(fact_5491_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5492_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5493_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5494_of__nat__le__0__iff,axiom,
    ! [Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int)
    <=> ( Mb = zero_zero_nat ) ) ).

% of_nat_le_0_iff
tff(fact_5495_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Mb)) = aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(semiri8010041392384452111omplex,Mb)) ).

% of_nat_Suc
tff(fact_5496_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb)) ).

% of_nat_Suc
tff(fact_5497_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb)) ).

% of_nat_Suc
tff(fact_5498_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) ).

% of_nat_Suc
tff(fact_5499_of__nat__Suc,axiom,
    ! [Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% of_nat_Suc
tff(fact_5500_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5501_bit__numeral__Bit0__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5502_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5503_bit__numeral__Bit1__Suc__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5504_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_5505_log__less__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_5506_log__less__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),A) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_5507_one__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_5508_log__less__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_5509_zero__less__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_5510_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_5511_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_5512_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_5513_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_5514_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_5515_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_5516_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5517_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5518_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5519_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_5520_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5521_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5522_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5523_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_5524_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_complex(semiri8010041392384452111omplex,Y) = aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5525_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Y) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5526_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_rat(semiri681578069525770553at_rat,Y) = aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5527_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_nat(semiri1316708129612266289at_nat,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5528_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,Xb: num,Nb: nat] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Y) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) )
    <=> ( Y = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_5529_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_complex(power_power_complex(numera6690914467698888265omplex(Xb)),Nb) = aa_nat_complex(semiri8010041392384452111omplex,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5530_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb) = aa_nat_real(semiri5074537144036343181t_real,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5531_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb) = aa_nat_rat(semiri681578069525770553at_rat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5532_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_nat_nat(semiri1316708129612266289at_nat,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5533_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: nat] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = aa_nat_int(semiri1314217659103216013at_int,Y) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_5534_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5535_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5536_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5537_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B2: nat,W: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)),Xb) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_5538_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5539_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5540_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5541_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [Xb: nat,B2: nat,W: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,B2)),W))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),W)) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_5542_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_5543_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_5544_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,Mb: 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,Mb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Nb)),Mb) ) ).

% numeral_le_real_of_nat_iff
tff(fact_5545_zero__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_5546_log__le__zero__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),zero_zero_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_5547_one__le__log__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_5548_log__le__one__cancel__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),one_one_real)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),A) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_5549_log__le__cancel__iff,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y))
          <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_5550_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_5551_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_5552_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_5553_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_5554_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_5555_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_5556_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5557_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(power_power_rat(aa_nat_rat(semiri681578069525770553at_rat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5558_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5559_of__nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(power_power_int(aa_nat_int(semiri1314217659103216013at_int,Xb)),Nb))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
        | ( Nb = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_5560_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_5561_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_5562_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_5563_log__pow__cancel,axiom,
    ! [A: real,B2: 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),B2)) = aa_nat_real(semiri5074537144036343181t_real,B2) ) ) ) ).

% log_pow_cancel
tff(fact_5564_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_5565_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_5566_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),semiri4939895301339042750nteger(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_5567_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_5568_even__of__nat,axiom,
    ! [Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ) ).

% even_of_nat
tff(fact_5569_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5570_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5571_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5572_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_5573_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5574_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5575_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5576_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_5577_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5578_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb)),aa_nat_rat(semiri681578069525770553at_rat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5579_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),aa_nat_nat(semiri1316708129612266289at_nat,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5580_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,Nb: nat,Xb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb)),aa_nat_int(semiri1314217659103216013at_int,Xb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)),Xb) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_5581_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5582_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5583_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5584_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [Xb: nat,I: num,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,I)),Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,I)),Nb)) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_5585_bit__mod__2__iff,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(modulo364778990260209775nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),Nb)
    <=> ( ( Nb = zero_zero_nat )
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5586_bit__mod__2__iff,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Nb)
    <=> ( ( 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) ) ) ).

% bit_mod_2_iff
tff(fact_5587_bit__mod__2__iff,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),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))),A) ) ) ).

% bit_mod_2_iff
tff(fact_5588_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_num_int(numeral_numeral_int,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_5589_bit__numeral__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% bit_numeral_iff
tff(fact_5590_bit__of__nat__iff__bit,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_5591_bit__of__nat__iff__bit,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),Nb)
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb) ) ).

% bit_of_nat_iff_bit
tff(fact_5592_bit__disjunctive__add__iff,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),N2)
          | ~ aa_nat_o(bit_se1146084159140164899it_int(B2),N2) )
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | aa_nat_o(bit_se1146084159140164899it_int(B2),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5593_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2)
          | ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B2),N2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B2),Nb) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5594_real__arch__simple,axiom,
    ! [Xb: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% real_arch_simple
tff(fact_5595_real__arch__simple,axiom,
    ! [Xb: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% real_arch_simple
tff(fact_5596_reals__Archimedean2,axiom,
    ! [Xb: real] :
    ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,N2)) ).

% reals_Archimedean2
tff(fact_5597_reals__Archimedean2,axiom,
    ! [Xb: rat] :
    ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_nat_rat(semiri681578069525770553at_rat,N2)) ).

% reals_Archimedean2
tff(fact_5598_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Xb)),Y) = aa_complex_complex(times_times_complex(Y),aa_nat_complex(semiri8010041392384452111omplex,Xb)) ).

% mult_of_nat_commute
tff(fact_5599_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_nat_real(semiri5074537144036343181t_real,Xb)) ).

% mult_of_nat_commute
tff(fact_5600_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Xb)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),aa_nat_rat(semiri681578069525770553at_rat,Xb)) ).

% mult_of_nat_commute
tff(fact_5601_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Xb)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),aa_nat_nat(semiri1316708129612266289at_nat,Xb)) ).

% mult_of_nat_commute
tff(fact_5602_mult__of__nat__commute,axiom,
    ! [Xb: nat,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(semiri1314217659103216013at_int,Xb)) ).

% mult_of_nat_commute
tff(fact_5603_bit__and__iff,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se725231765392027082nd_int(A),B2)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(B2),Nb) ) ) ).

% bit_and_iff
tff(fact_5604_bit__and__iff,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B2),Nb) ) ) ).

% bit_and_iff
tff(fact_5605_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_5606_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% arctan_less_iff
tff(fact_5607_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone
tff(fact_5608_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% arctan_le_iff
tff(fact_5609_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) ) ).

% arctan_monotone'
tff(fact_5610_bit__unset__bit__iff,axiom,
    ! [Mb: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),A)),Nb)
    <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        & ( Mb != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_5611_bit__unset__bit__iff,axiom,
    ! [Mb: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        & ( Mb != Nb ) ) ) ).

% bit_unset_bit_iff
tff(fact_5612_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Mb)) ) ) ).

% less_log_of_power
tff(fact_5613_log__of__power__eq,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( ( aa_nat_real(semiri5074537144036343181t_real,Mb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ) ).

% log_of_power_eq
tff(fact_5614_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_int_real(ring_1_of_int_real,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_5615_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_int_rat(ring_1_of_int_rat,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_5616_of__nat__less__of__int__iff,axiom,
    ! [Nb: nat,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(ring_1_of_int_int,Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Xb) ) ).

% of_nat_less_of_int_iff
tff(fact_5617_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,Mb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb)),Mb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => 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,B2),Mb)) ) ) ).

% le_log_of_power
tff(fact_5618_log__base__pow,axiom,
    ! [A: real,Nb: nat,Xb: 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)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log_base_pow
tff(fact_5619_log__nat__power,axiom,
    ! [Xb: real,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ).

% log_nat_power
tff(fact_5620_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_5621_not__bit__1__Suc,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,Nb)) ).

% not_bit_1_Suc
tff(fact_5622_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(one_one_int),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5623_bit__1__iff,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),Nb)
    <=> ( Nb = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5624_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(bit_se1146084159140164899it_int(one_one_int),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_5625_bit__numeral__simps_I1_J,axiom,
    ! [Nb: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,Nb)) ).

% bit_numeral_simps(1)
tff(fact_5626_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% of_nat_0_le_iff
tff(fact_5627_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% of_nat_0_le_iff
tff(fact_5628_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_0_le_iff
tff(fact_5629_of__nat__0__le__iff,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_0_le_iff
tff(fact_5630_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),zero_zero_real) ).

% of_nat_less_0_iff
tff(fact_5631_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),zero_zero_rat) ).

% of_nat_less_0_iff
tff(fact_5632_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),zero_zero_nat) ).

% of_nat_less_0_iff
tff(fact_5633_of__nat__less__0__iff,axiom,
    ! [Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),zero_zero_int) ).

% of_nat_less_0_iff
tff(fact_5634_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se2923211474154528505it_int(Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_5635_bit__take__bit__iff,axiom,
    ! [Mb: nat,A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),A)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ) ).

% bit_take_bit_iff
tff(fact_5636_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb)) != zero_zero_complex ).

% of_nat_neq_0
tff(fact_5637_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)) != zero_zero_real ).

% of_nat_neq_0
tff(fact_5638_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb)) != zero_zero_rat ).

% of_nat_neq_0
tff(fact_5639_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb)) != zero_zero_nat ).

% of_nat_neq_0
tff(fact_5640_of__nat__neq__0,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) != zero_zero_int ).

% of_nat_neq_0
tff(fact_5641_bit__of__bool__iff,axiom,
    ! [B2: $o,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(zero_n356916108424825756nteger((B2))),Nb)
    <=> ( (B2)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5642_bit__of__bool__iff,axiom,
    ! [B2: $o,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_o_int(zero_n2684676970156552555ol_int,(B2))),Nb)
    <=> ( (B2)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5643_bit__of__bool__iff,axiom,
    ! [B2: $o,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,zero_n2687167440665602831ol_nat((B2))),Nb)
    <=> ( (B2)
        & ( Nb = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
tff(fact_5644_div__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% div_mult2_eq'
tff(fact_5645_div__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% div_mult2_eq'
tff(fact_5646_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5647_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5648_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5649_less__imp__of__nat__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% less_imp_of_nat_less
tff(fact_5650_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5651_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5652_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5653_of__nat__less__imp__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% of_nat_less_imp_less
tff(fact_5654_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,I)),aa_nat_real(semiri5074537144036343181t_real,J)) ) ).

% of_nat_mono
tff(fact_5655_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,I)),aa_nat_rat(semiri681578069525770553at_rat,J)) ) ).

% of_nat_mono
tff(fact_5656_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1316708129612266289at_nat,I)),aa_nat_nat(semiri1316708129612266289at_nat,J)) ) ).

% of_nat_mono
tff(fact_5657_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,I)),aa_nat_int(semiri1314217659103216013at_int,J)) ) ).

% of_nat_mono
tff(fact_5658_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_5659_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_5660_log__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(ln_ln_real,A)) ).

% log_def
tff(fact_5661_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,semiri4939895301339042750nteger(Mb)),semiri4939895301339042750nteger(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_5662_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_5663_of__nat__dvd__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),Nb) ) ).

% of_nat_dvd_iff
tff(fact_5664_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,Nb: nat] :
      ( ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
     => ( aa_int_int(bit_ri631733984087533419it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),A) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_5665_abs__zmult__eq__1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)) = one_one_int )
     => ( aa_int_int(abs_abs_int,Mb) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_5666_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : semiri4939895301339042750nteger(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = modulo364778990260209775nteger(semiri4939895301339042750nteger(Mb),semiri4939895301339042750nteger(Nb)) ).

% of_nat_mod
tff(fact_5667_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_mod
tff(fact_5668_of__nat__mod,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_mod
tff(fact_5669_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Y),Xb)
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,Xb)),aa_int_int(abs_abs_int,Y)) ) ) ).

% abs_div
tff(fact_5670_log2__of__power__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( Mb = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) )
     => ( aa_nat_real(semiri5074537144036343181t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ) ).

% log2_of_power_eq
tff(fact_5671_log__of__power__less,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_less
tff(fact_5672_take__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_nat_int(semiri1314217659103216013at_int,Mb)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)) ).

% take_bit_of_nat
tff(fact_5673_take__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),Mb)) ).

% take_bit_of_nat
tff(fact_5674_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_and_eq
tff(fact_5675_of__nat__and__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_and_eq
tff(fact_5676_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_nat(bit_se2002935070580805687sk_nat,Nb) ).

% of_nat_mask_eq
tff(fact_5677_of__nat__mask__eq,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(bit_se2002935070580805687sk_nat,Nb)) = aa_nat_int(bit_se2000444600071755411sk_int,Nb) ).

% of_nat_mask_eq
tff(fact_5678_log__of__power__le,axiom,
    ! [Mb: nat,B2: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),Nb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_of_power_le
tff(fact_5679_ex__less__of__nat__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_5680_ex__less__of__nat__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,N2)),Xb)) ) ).

% ex_less_of_nat_mult
tff(fact_5681_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ) ) ).

% of_nat_diff
tff(fact_5682_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% of_nat_diff
tff(fact_5683_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ) ) ).

% of_nat_diff
tff(fact_5684_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ) ) ).

% of_nat_diff
tff(fact_5685_of__nat__diff,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ) ).

% of_nat_diff
tff(fact_5686_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int)),Nb)
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),Nb) ) ).

% bit_not_int_iff'
tff(fact_5687_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: complex] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),Xb)) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_5688_exp__of__nat__mult,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat_mult
tff(fact_5689_exp__of__nat2__mult,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_5690_exp__of__nat2__mult,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),Nb) ).

% exp_of_nat2_mult
tff(fact_5691_zabs__def,axiom,
    ! [I: int] :
      aa_int_int(abs_abs_int,I) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int),aa_int_int(uminus_uminus_int,I),I) ).

% zabs_def
tff(fact_5692_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ! [Y4: real] :
        ? [N2: 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,N2)),Xb)) ) ).

% reals_Archimedean3
tff(fact_5693_log__ln,axiom,
    ! [Xb: real] : aa_real_real(ln_ln_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(exp_real,one_one_real)),Xb) ).

% log_ln
tff(fact_5694_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))) ).

% real_of_nat_div4
tff(fact_5695_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))),aa_int_int(abs_abs_int,L)) ) ).

% abs_mod_less
tff(fact_5696_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_5697_less__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% less_log2_of_power
tff(fact_5698_le__log2__of__power,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),Mb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))) ) ).

% le_log2_of_power
tff(fact_5699_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_se2159334234014336723it_int(Nb),A) = aa_int_int(aa_nat_fun_int_int($ite(aa_nat_o(bit_se1146084159140164899it_int(A),Nb),bit_se4203085406695923979it_int,bit_se7879613467334960850it_int),Nb),A) ).

% flip_bit_eq_if
tff(fact_5700_flip__bit__eq__if,axiom,
    ! [Nb: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb),bit_se4205575877204974255it_nat,bit_se7882103937844011126it_nat),Nb),A) ).

% flip_bit_eq_if
tff(fact_5701_log2__of__power__less,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_less
tff(fact_5702_log__base__change,axiom,
    ! [A: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),B2)) ) ) ) ).

% log_base_change
tff(fact_5703_mod__mult2__eq_H,axiom,
    ! [A: code_integer,Mb: nat,Nb: nat] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,semiri4939895301339042750nteger(Mb)),semiri4939895301339042750nteger(Nb))) = plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,semiri4939895301339042750nteger(Mb)),modulo364778990260209775nteger(divide6298287555418463151nteger(A,semiri4939895301339042750nteger(Mb)),semiri4939895301339042750nteger(Nb))),modulo364778990260209775nteger(A,semiri4939895301339042750nteger(Mb))) ).

% mod_mult2_eq'
tff(fact_5704_mod__mult2__eq_H,axiom,
    ! [A: nat,Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(semiri1316708129612266289at_nat,Nb)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Mb))) ).

% mod_mult2_eq'
tff(fact_5705_mod__mult2__eq_H,axiom,
    ! [A: int,Mb: nat,Nb: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_int(semiri1314217659103216013at_int,Nb)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% mod_mult2_eq'
tff(fact_5706_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,Mb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
     => ( 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(Mb),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5707_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_complex(semiri8010041392384452111omplex,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_5708_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_5709_field__char__0__class_Oof__nat__div,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)))),aa_nat_rat(semiri681578069525770553at_rat,Nb)) ).

% field_char_0_class.of_nat_div
tff(fact_5710_bit__concat__bit__iff,axiom,
    ! [Mb: nat,K: int,L: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_concat_bit(Mb,K),L)),Nb)
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
          & aa_nat_o(bit_se1146084159140164899it_int(K),Nb) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
          & aa_nat_o(bit_se1146084159140164899it_int(L),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_5711_nat__less__real__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,Mb)) ) ).

% nat_less_real_le
tff(fact_5712_nat__le__real__less,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),one_one_real)) ) ).

% nat_le_real_less
tff(fact_5713_zdvd__mult__cancel1,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,Mb),Nb)),Mb)
      <=> ( aa_int_int(abs_abs_int,Nb) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_5714_log2__of__power__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ).

% log2_of_power_le
tff(fact_5715_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),aa_nat_real(semiri5074537144036343181t_real,D)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),D))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),D))),aa_nat_real(semiri5074537144036343181t_real,D))) ).

% real_of_nat_div_aux
tff(fact_5716_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_5717_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) = zero_zero_int )
     => ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5718_exp__eq__0__imp__not__bit,axiom,
    ! [Nb: nat,A: 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_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5719_bit__Suc,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),Nb) ) ).

% bit_Suc
tff(fact_5720_bit__Suc,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),Nb) ) ).

% bit_Suc
tff(fact_5721_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
      <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_5722_stable__imp__bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A )
     => ( aa_nat_o(bit_se1146084159140164899it_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) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_5723_stable__imp__bit__iff__odd,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_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_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) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_5724_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N2: nat] :
          ( aa_nat_o(bit_se9216721137139052372nteger(A),N2)
        <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) )
     => ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5725_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N2: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(A),N2)
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5726_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N2)
        <=> ~ 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,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = A ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5727_log__mult,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_5728_nat__approx__posE,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
     => ~ ! [N2: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),E) ) ).

% nat_approx_posE
tff(fact_5729_nat__approx__posE,axiom,
    ! [E: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),E)
     => ~ ! [N2: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),E) ) ).

% nat_approx_posE
tff(fact_5730_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_5731_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_5732_of__nat__less__two__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% of_nat_less_two_power
tff(fact_5733_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),M4)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),M4)
              <=> aa_nat_o(bit_se1146084159140164899it_int(K),N2) ) )
         => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
             => ( aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N2),one_one_nat))
              <=> ~ aa_nat_o(bit_se1146084159140164899it_int(K),N2) ) ) ) ).

% int_bit_bound
tff(fact_5734_log__divide,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
           => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),aa_real_real(aa_rea7340605660300375871l_real(log,A),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_5735_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_5736_inverse__of__nat__le,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( ( Nb != zero_zero_nat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_rat_rat(divide_divide_rat(one_one_rat),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ) ) ).

% inverse_of_nat_le
tff(fact_5737_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,aa_complex_complex(divide1717551699836669952omplex(Xb),aa_nat_complex(semiri8010041392384452111omplex,Nb)))),Nb) = aa_complex_complex(exp_complex,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_5738_exp__divide__power__eq,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb) = aa_real_real(exp_real,Xb) ) ) ).

% exp_divide_power_eq
tff(fact_5739_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_abs_add_iff
tff(fact_5740_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(aa_int_fun_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(aa_int_fun_int_int(plus_plus_int,K),L)) ) ).

% even_add_abs_iff
tff(fact_5741_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),C)
       => ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M3)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,M3)),Xb)),C) )
         => ( Xb = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_5742_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_5743_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,Xb))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),Xb)))),one_one_real) ).

% real_of_nat_div3
tff(fact_5744_ln__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_realpow
tff(fact_5745_bit__iff__odd,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb))) ) ).

% bit_iff_odd
tff(fact_5746_bit__iff__odd,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_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))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))) ) ).

% bit_iff_odd
tff(fact_5747_bit__iff__odd,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_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))),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))) ) ).

% bit_iff_odd
tff(fact_5748_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,Nb: nat] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) = zero_zero_int )
    <=> ~ aa_nat_o(bit_se1146084159140164899it_int(A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5749_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_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)) = zero_zero_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5750_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
         => ( ( B2 != one_one_real )
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_real_real(ln_ln_real,A))),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_5751_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(aa_int_fun_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_5752_nat__intermed__int__val,axiom,
    ! [Mb: nat,Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,Mb)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,Nb))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_5753_decr__lemma,axiom,
    ! [D: int,Xb: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Zb))),one_one_int)),D))),Zb) ) ).

% decr_lemma
tff(fact_5754_incr__lemma,axiom,
    ! [D: int,Zb: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Zb))),one_one_int)),D))) ) ).

% incr_lemma
tff(fact_5755_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)),one_one_real)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_5756_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_5757_even__bit__succ__iff,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(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(one_one_Code_integer,A)),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5758_even__bit__succ__iff,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(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5759_even__bit__succ__iff,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(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),A)),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5760_odd__bit__iff__bit__pred,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(bit_se9216721137139052372nteger(A),Nb)
      <=> ( aa_nat_o(bit_se9216721137139052372nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5761_odd__bit__iff__bit__pred,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(bit_se1146084159140164899it_int(A),Nb)
      <=> ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5762_odd__bit__iff__bit__pred,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(bit_se1148574629649215175it_nat,A),Nb)
      <=> ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),one_one_nat)),Nb)
          | ( Nb = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5763_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),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))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_5764_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B2: code_integer,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(plus_p5714425477246183910nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2))),Nb)
      <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),B2)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5765_bit__sum__mult__2__cases,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se1146084159140164899it_int(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2))),Nb)
      <=> $ite(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),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5766_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2))),Nb)
      <=> $ite(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),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),B2)),Nb)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5767_bit__rec,axiom,
    ! [A: code_integer,Nb: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),Nb)
    <=> $ite(Nb = zero_zero_nat,~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_5768_bit__rec,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> $ite(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),aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_5769_bit__rec,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> $ite(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),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_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_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ).

% bit_rec
tff(fact_5770_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),Nb)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),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))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),Nb)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_5771_set__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se7879613467334960850it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_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_5772_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,K),aa_int_int(aa_int_fun_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_5773_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(arctan,Xb)),aa_real_real(arctan,Y)) = aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_5774_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)),aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(bit_se1146084159140164899it_int(K),Nb)))),aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_5775_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Xb)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_5776_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)))),Nb)),aa_real_real(exp_real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_5777_lemma__termdiff3,axiom,
    ! [H: real,Zb: real,K5: real,Nb: nat] :
      ( ( H != zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Zb)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),H)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),Nb))),H)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(real_V7735802525324610683m_real,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5778_lemma__termdiff3,axiom,
    ! [H: complex,Zb: complex,K5: real,Nb: nat] :
      ( ( H != zero_zero_complex )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),K5)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Zb),H))),K5)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(plus_plus_complex(Zb),H)),Nb)),aa_nat_complex(power_power_complex(Zb),Nb))),H)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(power_power_complex(Zb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,K5),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_complex_real(real_V1022390504157884413omplex,H))) ) ) ) ).

% lemma_termdiff3
tff(fact_5779_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(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_5780_pochhammer__double,axiom,
    ! [Zb: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Zb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_complex(comm_s2602460028002588243omplex(Zb),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Zb),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_5781_pochhammer__double,axiom,
    ! [Zb: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Zb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_5782_pochhammer__double,axiom,
    ! [Zb: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Zb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),aa_nat_rat(comm_s4028243227959126397er_rat(Zb),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),Nb)) ).

% pochhammer_double
tff(fact_5783_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_5784_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(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),one_one_int) ) ) ).

% floor_log2_div2
tff(fact_5785_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_5786_of__int__floor__cancel,axiom,
    ! [Xb: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_floor_cancel
tff(fact_5787_of__int__floor__cancel,axiom,
    ! [Xb: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_floor_cancel
tff(fact_5788_of__int__ceiling__cancel,axiom,
    ! [Xb: rat] :
      ( ( aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_rat(ring_1_of_int_rat,N4) ) ).

% of_int_ceiling_cancel
tff(fact_5789_of__int__ceiling__cancel,axiom,
    ! [Xb: real] :
      ( ( aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb)) = Xb )
    <=> ? [N4: int] : Xb = aa_int_real(ring_1_of_int_real,N4) ) ).

% of_int_ceiling_cancel
tff(fact_5790_int__eq__iff__numeral,axiom,
    ! [Mb: nat,V: num] :
      ( ( aa_nat_int(semiri1314217659103216013at_int,Mb) = aa_num_int(numeral_numeral_int,V) )
    <=> ( Mb = aa_num_nat(numeral_numeral_nat,V) ) ) ).

% int_eq_iff_numeral
tff(fact_5791_floor__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_5792_floor__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% floor_numeral
tff(fact_5793_floor__one,axiom,
    aa_real_int(archim6058952711729229775r_real,one_one_real) = one_one_int ).

% floor_one
tff(fact_5794_floor__one,axiom,
    aa_rat_int(archim3151403230148437115or_rat,one_one_rat) = one_one_int ).

% floor_one
tff(fact_5795_ceiling__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_num_real(numeral_numeral_real,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_5796_ceiling__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_num_rat(numeral_numeral_rat,V)) = aa_num_int(numeral_numeral_int,V) ).

% ceiling_numeral
tff(fact_5797_pochhammer__0,axiom,
    ! [A: complex] : aa_nat_complex(comm_s2602460028002588243omplex(A),zero_zero_nat) = one_one_complex ).

% pochhammer_0
tff(fact_5798_pochhammer__0,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),zero_zero_nat) = one_one_real ).

% pochhammer_0
tff(fact_5799_pochhammer__0,axiom,
    ! [A: rat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),zero_zero_nat) = one_one_rat ).

% pochhammer_0
tff(fact_5800_pochhammer__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),zero_zero_nat) = one_one_nat ).

% pochhammer_0
tff(fact_5801_pochhammer__0,axiom,
    ! [A: int] : aa_nat_int(comm_s4660882817536571857er_int(A),zero_zero_nat) = one_one_int ).

% pochhammer_0
tff(fact_5802_ceiling__one,axiom,
    aa_rat_int(archim2889992004027027881ng_rat,one_one_rat) = one_one_int ).

% ceiling_one
tff(fact_5803_ceiling__one,axiom,
    aa_real_int(archim7802044766580827645g_real,one_one_real) = one_one_int ).

% ceiling_one
tff(fact_5804_negative__zless,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)))),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% negative_zless
tff(fact_5805_ceiling__add__of__int,axiom,
    ! [Xb: rat,Zb: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_int_rat(ring_1_of_int_rat,Zb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Zb) ).

% ceiling_add_of_int
tff(fact_5806_ceiling__add__of__int,axiom,
    ! [Xb: real,Zb: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Zb))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Zb) ).

% ceiling_add_of_int
tff(fact_5807_floor__diff__of__int,axiom,
    ! [Xb: real,Zb: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Zb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Zb) ).

% floor_diff_of_int
tff(fact_5808_floor__diff__of__int,axiom,
    ! [Xb: rat,Zb: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Zb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Zb) ).

% floor_diff_of_int
tff(fact_5809_ceiling__diff__of__int,axiom,
    ! [Xb: rat,Zb: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,Zb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Zb) ).

% ceiling_diff_of_int
tff(fact_5810_ceiling__diff__of__int,axiom,
    ! [Xb: real,Zb: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,Zb))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),Zb) ).

% ceiling_diff_of_int
tff(fact_5811_zero__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_floor
tff(fact_5812_zero__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb) ) ).

% zero_le_floor
tff(fact_5813_floor__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% floor_less_zero
tff(fact_5814_floor__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),zero_zero_rat) ) ).

% floor_less_zero
tff(fact_5815_numeral__le__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,V)),Xb) ) ).

% numeral_le_floor
tff(fact_5816_numeral__le__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).

% numeral_le_floor
tff(fact_5817_zero__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% zero_less_floor
tff(fact_5818_zero__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% zero_less_floor
tff(fact_5819_floor__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_le_zero
tff(fact_5820_floor__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_le_zero
tff(fact_5821_ceiling__le__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_le_zero
tff(fact_5822_ceiling__le__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_le_zero
tff(fact_5823_floor__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,V)) ) ).

% floor_less_numeral
tff(fact_5824_floor__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).

% floor_less_numeral
tff(fact_5825_zero__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% zero_less_ceiling
tff(fact_5826_zero__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% zero_less_ceiling
tff(fact_5827_one__le__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ).

% one_le_floor
tff(fact_5828_one__le__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ).

% one_le_floor
tff(fact_5829_ceiling__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_num_real(numeral_numeral_real,V)) ) ).

% ceiling_le_numeral
tff(fact_5830_ceiling__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_num_rat(numeral_numeral_rat,V)) ) ).

% ceiling_le_numeral
tff(fact_5831_floor__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ).

% floor_less_one
tff(fact_5832_floor__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ).

% floor_less_one
tff(fact_5833_ceiling__less__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% ceiling_less_one
tff(fact_5834_ceiling__less__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat) ) ).

% ceiling_less_one
tff(fact_5835_one__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb) ) ).

% one_le_ceiling
tff(fact_5836_one__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% one_le_ceiling
tff(fact_5837_numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(numeral_numeral_real,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_5838_numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(numeral_numeral_rat,V)),Xb) ) ).

% numeral_less_ceiling
tff(fact_5839_floor__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_5840_floor__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% floor_neg_numeral
tff(fact_5841_ceiling__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ).

% ceiling_le_one
tff(fact_5842_ceiling__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ).

% ceiling_le_one
tff(fact_5843_one__less__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ).

% one_less_ceiling
tff(fact_5844_one__less__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ).

% one_less_ceiling
tff(fact_5845_ceiling__add__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5846_ceiling__add__numeral,axiom,
    ! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_add_numeral
tff(fact_5847_floor__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_5848_floor__diff__numeral,axiom,
    ! [Xb: rat,V: num] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% floor_diff_numeral
tff(fact_5849_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_5850_ceiling__neg__numeral,axiom,
    ! [V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)) ).

% ceiling_neg_numeral
tff(fact_5851_ceiling__add__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_5852_ceiling__add__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_add_one
tff(fact_5853_floor__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) ).

% floor_diff_one
tff(fact_5854_floor__diff__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int) ).

% floor_diff_one
tff(fact_5855_ceiling__diff__numeral,axiom,
    ! [Xb: real,V: num] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_num_real(numeral_numeral_real,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5856_ceiling__diff__numeral,axiom,
    ! [Xb: rat,V: num] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),aa_num_rat(numeral_numeral_rat,V))) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V)) ).

% ceiling_diff_numeral
tff(fact_5857_ceiling__diff__one,axiom,
    ! [Xb: rat] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(minus_minus_rat(Xb),one_one_rat)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_5858_ceiling__diff__one,axiom,
    ! [Xb: real] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),one_one_real)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),one_one_int) ).

% ceiling_diff_one
tff(fact_5859_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_5860_floor__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% floor_numeral_power
tff(fact_5861_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_real_int(archim7802044766580827645g_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_5862_ceiling__numeral__power,axiom,
    ! [Xb: num,Nb: nat] : aa_rat_int(archim2889992004027027881ng_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,Xb)),Nb)) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ).

% ceiling_numeral_power
tff(fact_5863_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B2: 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,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_divide_eq_div_numeral
tff(fact_5864_ceiling__less__zero,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),zero_zero_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% ceiling_less_zero
tff(fact_5865_ceiling__less__zero,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),zero_zero_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% ceiling_less_zero
tff(fact_5866_zero__le__ceiling,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb) ) ).

% zero_le_ceiling
tff(fact_5867_zero__le__ceiling,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)),Xb) ) ).

% zero_le_ceiling
tff(fact_5868_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B2: 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,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_5869_numeral__less__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).

% numeral_less_floor
tff(fact_5870_numeral__less__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).

% numeral_less_floor
tff(fact_5871_floor__le__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% floor_le_numeral
tff(fact_5872_floor__le__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% floor_le_numeral
tff(fact_5873_one__less__floor,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_5874_one__less__floor,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),Xb) ) ).

% one_less_floor
tff(fact_5875_floor__le__one,axiom,
    ! [Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_5876_floor__le__one,axiom,
    ! [Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))) ) ).

% floor_le_one
tff(fact_5877_ceiling__less__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)) ) ).

% ceiling_less_numeral
tff(fact_5878_ceiling__less__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_num_int(numeral_numeral_int,V))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)) ) ).

% ceiling_less_numeral
tff(fact_5879_numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_num_real(numeral_numeral_real,V)),one_one_real)),Xb) ) ).

% numeral_le_ceiling
tff(fact_5880_numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,V)),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_num_rat(numeral_numeral_rat,V)),one_one_rat)),Xb) ) ).

% numeral_le_ceiling
tff(fact_5881_neg__numeral__le__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).

% neg_numeral_le_floor
tff(fact_5882_neg__numeral__le__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).

% neg_numeral_le_floor
tff(fact_5883_floor__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% floor_less_neg_numeral
tff(fact_5884_floor__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% floor_less_neg_numeral
tff(fact_5885_ceiling__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5886_ceiling__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))) ) ).

% ceiling_le_neg_numeral
tff(fact_5887_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_5888_neg__numeral__less__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),Xb) ) ).

% neg_numeral_less_ceiling
tff(fact_5889_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_num_int(numeral_numeral_int,B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_5890_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: 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,B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,A))),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_5891_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B2: 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,B2)))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,A)),aa_num_int(numeral_numeral_int,B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_5892_neg__numeral__less__floor,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).

% neg_numeral_less_floor
tff(fact_5893_neg__numeral__less__floor,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).

% neg_numeral_less_floor
tff(fact_5894_floor__le__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% floor_le_neg_numeral
tff(fact_5895_floor__le__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% floor_le_neg_numeral
tff(fact_5896_ceiling__less__neg__numeral,axiom,
    ! [Xb: real,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)) ) ).

% ceiling_less_neg_numeral
tff(fact_5897_ceiling__less__neg__numeral,axiom,
    ! [Xb: rat,V: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V)))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)) ) ).

% ceiling_less_neg_numeral
tff(fact_5898_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,V))),one_one_real)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_5899_neg__numeral__le__ceiling,axiom,
    ! [V: num,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,V))),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,V))),one_one_rat)),Xb) ) ).

% neg_numeral_le_ceiling
tff(fact_5900_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: 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,B2)))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_5901_int__if,axiom,
    ! [P: $o,A: nat,B2: nat] :
      aa_nat_int(semiri1314217659103216013at_int,
        $ite((P),A,B2)) = $ite((P),aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_if
tff(fact_5902_nat__int__comparison_I1_J,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( aa_nat_int(semiri1314217659103216013at_int,A) = aa_nat_int(semiri1314217659103216013at_int,B2) ) ) ).

% nat_int_comparison(1)
tff(fact_5903_int__diff__cases,axiom,
    ! [Zb: int] :
      ~ ! [M3: nat,N2: nat] : Zb != aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,M3)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_diff_cases
tff(fact_5904_ceiling__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim7802044766580827645g_real,Xb) = $ite(Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_5905_ceiling__altdef,axiom,
    ! [Xb: rat] :
      aa_rat_int(archim2889992004027027881ng_rat,Xb) = $ite(Xb = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Xb),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int)) ).

% ceiling_altdef
tff(fact_5906_ceiling__diff__floor__le__1,axiom,
    ! [Xb: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim6058952711729229775r_real,Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_5907_ceiling__diff__floor__le__1,axiom,
    ! [Xb: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Xb))),one_one_int) ).

% ceiling_diff_floor_le_1
tff(fact_5908_floor__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ).

% floor_mono
tff(fact_5909_floor__mono,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ).

% floor_mono
tff(fact_5910_of__int__floor__le,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb) ).

% of_int_floor_le
tff(fact_5911_of__int__floor__le,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb))),Xb) ).

% of_int_floor_le
tff(fact_5912_floor__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% floor_less_cancel
tff(fact_5913_floor__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% floor_less_cancel
tff(fact_5914_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_5915_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_5916_ceiling__mono,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Y)),aa_real_int(archim7802044766580827645g_real,Xb)) ) ).

% ceiling_mono
tff(fact_5917_ceiling__mono,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Y)),aa_rat_int(archim2889992004027027881ng_rat,Xb)) ) ).

% ceiling_mono
tff(fact_5918_le__of__int__ceiling,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ).

% le_of_int_ceiling
tff(fact_5919_le__of__int__ceiling,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ).

% le_of_int_ceiling
tff(fact_5920_ceiling__less__cancel,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_5921_ceiling__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% ceiling_less_cancel
tff(fact_5922_int__ops_I1_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,zero_zero_nat) = zero_zero_int ).

% int_ops(1)
tff(fact_5923_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_5924_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_5925_pochhammer__pos,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_5926_pochhammer__pos,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_5927_pochhammer__pos,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_5928_pochhammer__pos,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_pos
tff(fact_5929_nat__int__comparison_I2_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(2)
tff(fact_5930_int__of__nat__induct,axiom,
    ! [P: fun_int_o,Zb: int] :
      ( ! [N2: nat] : aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,N2))
     => ( ! [N2: nat] : aa_int_o(P,aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))))
       => aa_int_o(P,Zb) ) ) ).

% int_of_nat_induct
tff(fact_5931_int__cases,axiom,
    ! [Zb: int] :
      ( ! [N2: nat] : Zb != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] : Zb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% int_cases
tff(fact_5932_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_5933_pi__gt__zero,axiom,
    aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),pi) ).

% pi_gt_zero
tff(fact_5934_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_5935_nat__int__comparison_I3_J,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ) ).

% nat_int_comparison(3)
tff(fact_5936_zle__int,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% zle_int
tff(fact_5937_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(B2),A))),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% complex_mod_triangle_ineq2
tff(fact_5938_pochhammer__eq__0__mono,axiom,
    ! [A: real,Nb: nat,Mb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5939_pochhammer__eq__0__mono,axiom,
    ! [A: rat,Nb: nat,Mb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_5940_pochhammer__neq__0__mono,axiom,
    ! [A: real,Mb: nat,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Mb) != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5941_pochhammer__neq__0__mono,axiom,
    ! [A: rat,Mb: nat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Mb) != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
       => ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_5942_int__ops_I5_J,axiom,
    ! [A: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(5)
tff(fact_5943_int__plus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)) ).

% int_plus
tff(fact_5944_zadd__int__left,axiom,
    ! [Mb: nat,Nb: nat,Zb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),Zb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))),Zb) ).

% zadd_int_left
tff(fact_5945_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_5946_int__ops_I7_J,axiom,
    ! [A: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% int_ops(7)
tff(fact_5947_zle__iff__zadd,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb)
    <=> ? [N4: nat] : Zb = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,N4)) ) ).

% zle_iff_zadd
tff(fact_5948_not__int__zless__negative,axiom,
    ! [Nb: nat,Mb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Mb))) ).

% not_int_zless_negative
tff(fact_5949_zdiv__int,axiom,
    ! [A: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zdiv_int
tff(fact_5950_zmod__int,axiom,
    ! [A: nat,B2: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% zmod_int
tff(fact_5951_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_5952_floor__divide__of__nat__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri5074537144036343181t_real,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_5953_floor__divide__of__nat__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri681578069525770553at_rat,Nb))) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)) ).

% floor_divide_of_nat_eq
tff(fact_5954_le__floor__iff,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Zb)),Xb) ) ).

% le_floor_iff
tff(fact_5955_le__floor__iff,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Zb)),Xb) ) ).

% le_floor_iff
tff(fact_5956_floor__less__iff,axiom,
    ! [Xb: real,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim6058952711729229775r_real,Xb)),Zb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,Zb)) ) ).

% floor_less_iff
tff(fact_5957_floor__less__iff,axiom,
    ! [Xb: rat,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Zb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,Zb)) ) ).

% floor_less_iff
tff(fact_5958_le__floor__add,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))) ).

% le_floor_add
tff(fact_5959_le__floor__add,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y))) ).

% le_floor_add
tff(fact_5960_int__add__floor,axiom,
    ! [Zb: int,Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),aa_real_int(archim6058952711729229775r_real,Xb)) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Zb)),Xb)) ).

% int_add_floor
tff(fact_5961_int__add__floor,axiom,
    ! [Zb: int,Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),aa_rat_int(archim3151403230148437115or_rat,Xb)) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Zb)),Xb)) ).

% int_add_floor
tff(fact_5962_floor__add__int,axiom,
    ! [Xb: real,Zb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),Zb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_int_real(ring_1_of_int_real,Zb))) ).

% floor_add_int
tff(fact_5963_floor__add__int,axiom,
    ! [Xb: rat,Zb: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Zb) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_int_rat(ring_1_of_int_rat,Zb))) ).

% floor_add_int
tff(fact_5964_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,K)),aa_int_real(ring_1_of_int_real,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_5965_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] : aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) ).

% floor_divide_of_int_eq
tff(fact_5966_floor__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb = aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb)) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_int(power_power_int(aa_real_int(archim6058952711729229775r_real,Xb)),Nb) ) ) ).

% floor_power
tff(fact_5967_floor__power,axiom,
    ! [Xb: rat,Nb: nat] :
      ( ( Xb = aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb)) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_nat_rat(power_power_rat(Xb),Nb)) = aa_nat_int(power_power_int(aa_rat_int(archim3151403230148437115or_rat,Xb)),Nb) ) ) ).

% floor_power
tff(fact_5968_ceiling__le,axiom,
    ! [Xb: real,A: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),A) ) ).

% ceiling_le
tff(fact_5969_ceiling__le,axiom,
    ! [Xb: rat,A: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),A) ) ).

% ceiling_le
tff(fact_5970_ceiling__le__iff,axiom,
    ! [Xb: real,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,Xb)),Zb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Zb)) ) ).

% ceiling_le_iff
tff(fact_5971_ceiling__le__iff,axiom,
    ! [Xb: rat,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Zb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Zb)) ) ).

% ceiling_le_iff
tff(fact_5972_less__ceiling__iff,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Zb)),Xb) ) ).

% less_ceiling_iff
tff(fact_5973_less__ceiling__iff,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Zb)),Xb) ) ).

% less_ceiling_iff
tff(fact_5974_ceiling__add__le,axiom,
    ! [Xb: rat,Y: rat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Y))) ).

% ceiling_add_le
tff(fact_5975_ceiling__add__le,axiom,
    ! [Xb: real,Y: real] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim7802044766580827645g_real,Xb)),aa_real_int(archim7802044766580827645g_real,Y))) ).

% ceiling_add_le
tff(fact_5976_pochhammer__nonneg,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5977_pochhammer__nonneg,axiom,
    ! [Xb: rat,Nb: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(comm_s4028243227959126397er_rat(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5978_pochhammer__nonneg,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5979_pochhammer__nonneg,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(comm_s4660882817536571857er_int(Xb),Nb)) ) ).

% pochhammer_nonneg
tff(fact_5980_int__cases4,axiom,
    ! [Mb: int] :
      ( ! [N2: nat] : Mb != aa_nat_int(semiri1314217659103216013at_int,N2)
     => ~ ! [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
           => ( Mb != aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) ) ) ) ).

% int_cases4
tff(fact_5981_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(zero_zero_complex),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,zero_zero_complex) ).

% pochhammer_0_left
tff(fact_5982_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,zero_zero_real),Nb) = $ite(Nb = zero_zero_nat,one_one_real,zero_zero_real) ).

% pochhammer_0_left
tff(fact_5983_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(zero_zero_rat),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% pochhammer_0_left
tff(fact_5984_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,zero_zero_nat),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,zero_zero_nat) ).

% pochhammer_0_left
tff(fact_5985_pochhammer__0__left,axiom,
    ! [Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(zero_zero_int),Nb) = $ite(Nb = zero_zero_nat,one_one_int,zero_zero_int) ).

% pochhammer_0_left
tff(fact_5986_int__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ).

% int_Suc
tff(fact_5987_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_5988_zless__iff__Suc__zadd,axiom,
    ! [W: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb)
    <=> ? [N4: nat] : Zb = aa_int_int(aa_int_fun_int_int(plus_plus_int,W),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N4))) ) ).

% zless_iff_Suc_zadd
tff(fact_5989_norm__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Xb))),aa_real_real(exp_real,aa_real_real(real_V7735802525324610683m_real,Xb))) ).

% norm_exp
tff(fact_5990_norm__exp,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Xb))),aa_real_real(exp_real,aa_complex_real(real_V1022390504157884413omplex,Xb))) ).

% norm_exp
tff(fact_5991_one__add__floor,axiom,
    ! [Xb: real] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% one_add_floor
tff(fact_5992_one__add__floor,axiom,
    ! [Xb: rat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) ).

% one_add_floor
tff(fact_5993_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),one_one_real)) ).

% of_int_ceiling_le_add_one
tff(fact_5994_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,R2),one_one_rat)) ).

% of_int_ceiling_le_add_one
tff(fact_5995_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,R2))),one_one_real)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_5996_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,R2))),one_one_rat)),R2) ).

% of_int_ceiling_diff_one_le
tff(fact_5997_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_5998_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_5999_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_6000_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_6001_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_6002_ceiling__divide__eq__div,axiom,
    ! [A: int,B2: int] : aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B2)) ).

% ceiling_divide_eq_div
tff(fact_6003_ceiling__divide__eq__div,axiom,
    ! [A: int,B2: int] : aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B2))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B2)) ).

% ceiling_divide_eq_div
tff(fact_6004_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & ( K = aa_nat_int(semiri1314217659103216013at_int,N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_6005_pos__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),K)
     => ~ ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% pos_int_cases
tff(fact_6006_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( ! [N2: nat] :
            ( ( K = aa_nat_int(semiri1314217659103216013at_int,N2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) )
       => ~ ! [N2: nat] :
              ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ) ).

% int_cases3
tff(fact_6007_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_6008_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_6009_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),J)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),I)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_6010_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_6011_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_6012_negD,axiom,
    ! [Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),zero_zero_int)
     => ? [N2: nat] : Xb = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,N2))) ) ).

% negD
tff(fact_6013_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_6014_pochhammer__rec,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),Nb)) ).

% pochhammer_rec
tff(fact_6015_pochhammer__rec,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),Nb)) ).

% pochhammer_rec
tff(fact_6016_pochhammer__rec,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),Nb)) ).

% pochhammer_rec
tff(fact_6017_pochhammer__rec,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),Nb)) ).

% pochhammer_rec
tff(fact_6018_pochhammer__rec,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),Nb)) ).

% pochhammer_rec
tff(fact_6019_pochhammer__Suc,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(A),Nb)),aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,Nb))) ).

% pochhammer_Suc
tff(fact_6020_pochhammer__Suc,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,Nb))) ).

% pochhammer_Suc
tff(fact_6021_pochhammer__Suc,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,Nb))) ).

% pochhammer_Suc
tff(fact_6022_pochhammer__Suc,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(semiri1316708129612266289at_nat,Nb))) ).

% pochhammer_Suc
tff(fact_6023_pochhammer__Suc,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(A),Nb)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% pochhammer_Suc
tff(fact_6024_pochhammer__rec_H,axiom,
    ! [Zb: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Zb),aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(Zb),aa_nat_complex(semiri8010041392384452111omplex,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(Zb),Nb)) ).

% pochhammer_rec'
tff(fact_6025_pochhammer__rec_H,axiom,
    ! [Zb: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),Nb)) ).

% pochhammer_rec'
tff(fact_6026_pochhammer__rec_H,axiom,
    ! [Zb: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Zb),aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_nat_rat(semiri681578069525770553at_rat,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(Zb),Nb)) ).

% pochhammer_rec'
tff(fact_6027_pochhammer__rec_H,axiom,
    ! [Zb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Zb),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),Nb)) ).

% pochhammer_rec'
tff(fact_6028_pochhammer__rec_H,axiom,
    ! [Zb: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Zb),aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),aa_nat_int(semiri1314217659103216013at_int,Nb))),aa_nat_int(comm_s4660882817536571857er_int(Zb),Nb)) ).

% pochhammer_rec'
tff(fact_6029_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(Nb)),K) = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6030_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6031_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6032_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6033_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [Nb: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_6034_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(Nb)),K) = zero_z3403309356797280102nteger )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6035_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) = zero_zero_complex )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6036_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) = zero_zero_real )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6037_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) = zero_zero_rat )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6038_pochhammer__of__nat__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) = zero_zero_int )
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),K) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_6039_pochhammer__eq__0__iff,axiom,
    ! [A: complex,Nb: nat] :
      ( ( aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = zero_zero_complex )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6040_pochhammer__eq__0__iff,axiom,
    ! [A: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = zero_zero_real )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6041_pochhammer__eq__0__iff,axiom,
    ! [A: rat,Nb: nat] :
      ( ( aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = zero_zero_rat )
    <=> ? [K3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K3),Nb)
          & ( A = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K3)) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_6042_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(Nb)),K) != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6043_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),K) != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6044_pochhammer__of__nat__eq__0__lemma_H,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(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),K) != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6045_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),K) != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6046_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),K) != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_6047_int__ops_I6_J,axiom,
    ! [A: nat,B2: nat] :
      aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2))) ).

% int_ops(6)
tff(fact_6048_pochhammer__product_H,axiom,
    ! [Zb: complex,Nb: nat,Mb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(Zb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Zb),Nb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Zb),aa_nat_complex(semiri8010041392384452111omplex,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6049_pochhammer__product_H,axiom,
    ! [Zb: real,Nb: nat,Mb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),Nb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),aa_nat_real(semiri5074537144036343181t_real,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6050_pochhammer__product_H,axiom,
    ! [Zb: rat,Nb: nat,Mb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(Zb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Zb),Nb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_nat_rat(semiri681578069525770553at_rat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6051_pochhammer__product_H,axiom,
    ! [Zb: nat,Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Zb),aa_nat_nat(semiri1316708129612266289at_nat,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6052_pochhammer__product_H,axiom,
    ! [Zb: int,Nb: nat,Mb: nat] : aa_nat_int(comm_s4660882817536571857er_int(Zb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Zb),Nb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),aa_nat_int(semiri1314217659103216013at_int,Nb))),Mb)) ).

% pochhammer_product'
tff(fact_6053_floor__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim6058952711729229775r_real,Ta))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,I2)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Ta),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_6054_floor__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim3151403230148437115or_rat,Ta))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,I2)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Ta),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)) )
         => aa_int_o(P,I2) ) ) ).

% floor_split
tff(fact_6055_floor__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim6058952711729229775r_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,A)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)) ) ) ).

% floor_eq_iff
tff(fact_6056_floor__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( aa_rat_int(archim3151403230148437115or_rat,Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,A)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)) ) ) ).

% floor_eq_iff
tff(fact_6057_floor__unique,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Zb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Zb ) ) ) ).

% floor_unique
tff(fact_6058_floor__unique,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Zb)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat))
       => ( aa_rat_int(archim3151403230148437115or_rat,Xb) = Zb ) ) ) ).

% floor_unique
tff(fact_6059_le__mult__floor,axiom,
    ! [A: real,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2))),aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2))) ) ) ).

% le_mult_floor
tff(fact_6060_le__mult__floor,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2))),aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))) ) ) ).

% le_mult_floor
tff(fact_6061_less__floor__iff,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_real_int(archim6058952711729229775r_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)),Xb) ) ).

% less_floor_iff
tff(fact_6062_less__floor__iff,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Zb),aa_rat_int(archim3151403230148437115or_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)),Xb) ) ).

% less_floor_iff
tff(fact_6063_floor__le__iff,axiom,
    ! [Xb: real,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim6058952711729229775r_real,Xb)),Zb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)) ) ).

% floor_le_iff
tff(fact_6064_floor__le__iff,axiom,
    ! [Xb: rat,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),Zb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)) ) ).

% floor_le_iff
tff(fact_6065_binomial__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_mono
tff(fact_6066_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_6067_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_6068_binomial__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),K)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_6069_floor__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_6070_floor__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb))),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),one_one_int))) ) ).

% floor_correct
tff(fact_6071_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: real] :
      ( aa_int_o(P,aa_real_int(archim7802044766580827645g_real,Ta))
    <=> ! [I2: int] :
          ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,I2)),one_one_real)),Ta)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Ta),aa_int_real(ring_1_of_int_real,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_6072_ceiling__split,axiom,
    ! [P: fun_int_o,Ta: rat] :
      ( aa_int_o(P,aa_rat_int(archim2889992004027027881ng_rat,Ta))
    <=> ! [I2: int] :
          ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,I2)),one_one_rat)),Ta)
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Ta),aa_int_rat(ring_1_of_int_rat,I2)) )
         => aa_int_o(P,I2) ) ) ).

% ceiling_split
tff(fact_6073_ceiling__eq__iff,axiom,
    ! [Xb: real,A: int] :
      ( ( aa_real_int(archim7802044766580827645g_real,Xb) = A )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,A)),one_one_real)),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6074_ceiling__eq__iff,axiom,
    ! [Xb: rat,A: int] :
      ( ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = A )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,A)),one_one_rat)),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,A)) ) ) ).

% ceiling_eq_iff
tff(fact_6075_ceiling__unique,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,Zb))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = Zb ) ) ) ).

% ceiling_unique
tff(fact_6076_ceiling__unique,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,Zb))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = Zb ) ) ) ).

% ceiling_unique
tff(fact_6077_ceiling__correct,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))),one_one_real)),Xb)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,Xb))) ) ).

% ceiling_correct
tff(fact_6078_ceiling__correct,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))),one_one_rat)),Xb)
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,Xb))) ) ).

% ceiling_correct
tff(fact_6079_mult__ceiling__le,axiom,
    ! [A: real,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2))) ) ) ).

% mult_ceiling_le
tff(fact_6080_mult__ceiling__le,axiom,
    ! [A: rat,B2: 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),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2))) ) ) ).

% mult_ceiling_le
tff(fact_6081_floor__log__nat__eq__if,axiom,
    ! [B2: 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,B2),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,B2),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))),B2)
         => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_nat_int(semiri1314217659103216013at_int,Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_6082_ceiling__less__iff,axiom,
    ! [Xb: real,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(archim7802044766580827645g_real,Xb)),Zb)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)) ) ).

% ceiling_less_iff
tff(fact_6083_ceiling__less__iff,axiom,
    ! [Xb: rat,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(archim2889992004027027881ng_rat,Xb)),Zb)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)) ) ).

% ceiling_less_iff
tff(fact_6084_le__ceiling__iff,axiom,
    ! [Zb: int,Xb: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_rat_int(archim2889992004027027881ng_rat,Xb))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,Zb)),one_one_rat)),Xb) ) ).

% le_ceiling_iff
tff(fact_6085_le__ceiling__iff,axiom,
    ! [Zb: int,Xb: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),aa_real_int(archim7802044766580827645g_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,Zb)),one_one_real)),Xb) ) ).

% le_ceiling_iff
tff(fact_6086_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim6058952711729229775r_real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_6087_floor__divide__real__eq__div,axiom,
    ! [B2: int,A: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_int_real(ring_1_of_int_real,B2))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_real_int(archim6058952711729229775r_real,A)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_6088_neg__int__cases,axiom,
    ! [K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int)
     => ~ ! [N2: nat] :
            ( ( K = aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,N2)) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2) ) ) ).

% neg_int_cases
tff(fact_6089_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_6090_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_6091_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_6092_zdiff__int__split,axiom,
    ! [P: fun_int_o,Xb: nat,Y: nat] :
      ( aa_int_o(P,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Y)))
    <=> ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Xb)),aa_nat_int(semiri1314217659103216013at_int,Y))) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
         => aa_int_o(P,zero_zero_int) ) ) ) ).

% zdiff_int_split
tff(fact_6093_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: 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))),B2)
     => ( 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,B2)),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,B2),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,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_6094_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Zb: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_complex(comm_s2602460028002588243omplex(Zb),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Zb),Mb)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Zb),aa_nat_complex(semiri8010041392384452111omplex,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6095_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Zb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),Mb)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),aa_nat_real(semiri5074537144036343181t_real,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6096_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Zb: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_rat(comm_s4028243227959126397er_rat(Zb),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Zb),Mb)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_nat_rat(semiri681578069525770553at_rat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6097_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Zb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,Zb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Zb),aa_nat_nat(semiri1316708129612266289at_nat,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6098_pochhammer__product,axiom,
    ! [Mb: nat,Nb: nat,Zb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_int(comm_s4660882817536571857er_int(Zb),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(comm_s4660882817536571857er_int(Zb),Mb)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb))) ) ) ).

% pochhammer_product
tff(fact_6099_floor__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2)),P2) ) ).

% floor_divide_lower
tff(fact_6100_floor__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),Q2)),P2) ) ).

% floor_divide_lower
tff(fact_6101_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_6102_binomial__strict__mono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K6)),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)) ) ) ).

% binomial_strict_mono
tff(fact_6103_binomial__strict__antimono,axiom,
    ! [K: nat,K6: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),K6)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),K))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K6),Nb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K6)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_6104_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_6105_ceiling__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_6106_ceiling__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),Q2)) ) ).

% ceiling_divide_upper
tff(fact_6107_bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Mb),Nb)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ) ).

% bit_nat_def
tff(fact_6108_ceiling__log__nat__eq__if,axiom,
    ! [B2: 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,B2),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,B2),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))),B2)
         => ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,aa_nat_real(semiri5074537144036343181t_real,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_6109_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_6110_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_6111_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(R2,semiri4939895301339042750nteger(K))),comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),K)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,R2),one_one_Code_integer),K)) ).

% pochhammer_absorb_comp
tff(fact_6112_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(R2),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,R2)),K)) = aa_complex_complex(times_times_complex(R2),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(uminus1482373934393186551omplex,R2)),one_one_complex)),K)) ).

% pochhammer_absorb_comp
tff(fact_6113_pochhammer__absorb__comp,axiom,
    ! [R2: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,R2)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(uminus_uminus_real,R2)),one_one_real)),K)) ).

% pochhammer_absorb_comp
tff(fact_6114_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(R2),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,R2)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(uminus_uminus_rat,R2)),one_one_rat)),K)) ).

% pochhammer_absorb_comp
tff(fact_6115_pochhammer__absorb__comp,axiom,
    ! [R2: int,K: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_nat_int(semiri1314217659103216013at_int,K))),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,R2)),K)) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,R2)),one_one_int)),K)) ).

% pochhammer_absorb_comp
tff(fact_6116_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_6117_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: 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))),B2)
     => ( 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,B2)),aa_nat_real(semiri5074537144036343181t_real,K))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),one_one_int) )
        <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B2),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,B2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_6118_floor__divide__upper,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,P2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2)) ) ).

% floor_divide_upper
tff(fact_6119_floor__divide__upper,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),one_one_rat)),Q2)) ) ).

% floor_divide_upper
tff(fact_6120_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_6121_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_6122_round__def,axiom,
    ! [Xb: real] : aa_real_int(archim8280529875227126926d_real,Xb) = aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_6123_round__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim7778729529865785530nd_rat,Xb) = aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))) ).

% round_def
tff(fact_6124_ceiling__divide__lower,axiom,
    ! [Q2: real,P2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Q2)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,P2),Q2)))),one_one_real)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_6125_ceiling__divide__lower,axiom,
    ! [Q2: rat,P2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Q2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(divide_divide_rat(P2),Q2)))),one_one_rat)),Q2)),P2) ) ).

% ceiling_divide_lower
tff(fact_6126_ceiling__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_int_real(ring_1_of_int_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_int_real(ring_1_of_int_real,Nb)),one_one_real))
       => ( aa_real_int(archim7802044766580827645g_real,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6127_ceiling__eq,axiom,
    ! [Nb: int,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(ring_1_of_int_rat,Nb)),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(ring_1_of_int_rat,Nb)),one_one_rat))
       => ( aa_rat_int(archim2889992004027027881ng_rat,Xb) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int) ) ) ) ).

% ceiling_eq
tff(fact_6128_exp__bound__half,axiom,
    ! [Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Zb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_6129_exp__bound__half,axiom,
    ! [Zb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Zb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ).

% exp_bound_half
tff(fact_6130_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_6131_pochhammer__minus,axiom,
    ! [B2: code_integer,K: nat] : comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),K)),comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B2,semiri4939895301339042750nteger(K)),one_one_Code_integer),K)) ).

% pochhammer_minus
tff(fact_6132_pochhammer__minus,axiom,
    ! [B2: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B2)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B2),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% pochhammer_minus
tff(fact_6133_pochhammer__minus,axiom,
    ! [B2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B2)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% pochhammer_minus
tff(fact_6134_pochhammer__minus,axiom,
    ! [B2: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B2)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B2),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% pochhammer_minus
tff(fact_6135_pochhammer__minus,axiom,
    ! [B2: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B2)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K)) ).

% pochhammer_minus
tff(fact_6136_pochhammer__minus_H,axiom,
    ! [B2: code_integer,K: nat] : comm_s8582702949713902594nteger(plus_p5714425477246183910nteger(minus_8373710615458151222nteger(B2,semiri4939895301339042750nteger(K)),one_one_Code_integer),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),K)),comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B2),K)) ).

% pochhammer_minus'
tff(fact_6137_pochhammer__minus_H,axiom,
    ! [B2: complex,K: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(B2),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,B2)),K)) ).

% pochhammer_minus'
tff(fact_6138_pochhammer__minus_H,axiom,
    ! [B2: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,B2)),K)) ).

% pochhammer_minus'
tff(fact_6139_pochhammer__minus_H,axiom,
    ! [B2: rat,K: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(B2),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,B2)),K)) ).

% pochhammer_minus'
tff(fact_6140_pochhammer__minus_H,axiom,
    ! [B2: int,K: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),aa_nat_int(semiri1314217659103216013at_int,K))),one_one_int)),K) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),K)),aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,B2)),K)) ).

% pochhammer_minus'
tff(fact_6141_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_6142_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_6143_exp__bound__lemma,axiom,
    ! [Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(exp_real,Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(real_V7735802525324610683m_real,Zb)))) ) ).

% exp_bound_lemma
tff(fact_6144_exp__bound__lemma,axiom,
    ! [Zb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(exp_complex,Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(real_V1022390504157884413omplex,Zb)))) ) ).

% exp_bound_lemma
tff(fact_6145_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_6146_norm__divide__numeral,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_6147_norm__divide__numeral,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_divide_numeral
tff(fact_6148_norm__mult__numeral1,axiom,
    ! [W: num,A: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_real_real(real_V7735802525324610683m_real,A)) ).

% norm_mult_numeral1
tff(fact_6149_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(W)),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,W)),aa_complex_real(real_V1022390504157884413omplex,A)) ).

% norm_mult_numeral1
tff(fact_6150_norm__mult__numeral2,axiom,
    ! [A: real,W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_6151_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(A),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_num_real(numeral_numeral_real,W)) ).

% norm_mult_numeral2
tff(fact_6152_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_6153_norm__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_6154_norm__neg__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_num_real(numeral_numeral_real,W) ).

% norm_neg_numeral
tff(fact_6155_norm__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_real ) ) ).

% norm_le_zero_iff
tff(fact_6156_norm__le__zero__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real)
    <=> ( Xb = zero_zero_complex ) ) ).

% norm_le_zero_iff
tff(fact_6157_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_6158_binomial__n__n,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Nb) = one_one_nat ).

% binomial_n_n
tff(fact_6159_norm__one,axiom,
    aa_real_real(real_V7735802525324610683m_real,one_one_real) = one_one_real ).

% norm_one
tff(fact_6160_norm__one,axiom,
    aa_complex_real(real_V1022390504157884413omplex,one_one_complex) = one_one_real ).

% norm_one
tff(fact_6161_norm__numeral,axiom,
    ! [W: num] : aa_real_real(real_V7735802525324610683m_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_6162_norm__numeral,axiom,
    ! [W: num] : aa_complex_real(real_V1022390504157884413omplex,numera6690914467698888265omplex(W)) = aa_num_real(numeral_numeral_real,W) ).

% norm_numeral
tff(fact_6163_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_6164_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_6165_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_6166_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_6167_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_6168_zero__less__norm__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(real_V7735802525324610683m_real,Xb))
    <=> ( Xb != zero_zero_real ) ) ).

% zero_less_norm_iff
tff(fact_6169_zero__less__norm__iff,axiom,
    ! [Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb))
    <=> ( Xb != zero_zero_complex ) ) ).

% zero_less_norm_iff
tff(fact_6170_norm__minus__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) ).

% norm_minus_commute
tff(fact_6171_norm__minus__commute,axiom,
    ! [A: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),A)) ).

% norm_minus_commute
tff(fact_6172_choose__one,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),one_one_nat) = Nb ).

% choose_one
tff(fact_6173_norm__not__less__zero,axiom,
    ! [Xb: complex] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),zero_zero_real) ).

% norm_not_less_zero
tff(fact_6174_norm__ge__zero,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% norm_ge_zero
tff(fact_6175_norm__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y)) ).

% norm_mult
tff(fact_6176_norm__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y)) ).

% norm_mult
tff(fact_6177_norm__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2)) ).

% norm_divide
tff(fact_6178_norm__divide,axiom,
    ! [A: complex,B2: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2)) ).

% norm_divide
tff(fact_6179_norm__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb) ).

% norm_power
tff(fact_6180_norm__power,axiom,
    ! [Xb: complex,Nb: nat] : aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb) ).

% norm_power
tff(fact_6181_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_6182_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_6183_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_6184_binomial__symmetric,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(binomial,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_6185_choose__mult__lemma,axiom,
    ! [Mb: 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,Mb),R2)),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K))),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),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,Mb),R2)),K)),K)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),R2)),Mb)) ).

% choose_mult_lemma
tff(fact_6186_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_6187_norm__uminus__minus,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,Xb)),Y)) = aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ).

% norm_uminus_minus
tff(fact_6188_norm__uminus__minus,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,Xb)),Y)) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y)) ).

% norm_uminus_minus
tff(fact_6189_nonzero__norm__divide,axiom,
    ! [B2: real,A: real] :
      ( ( B2 != zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_6190_nonzero__norm__divide,axiom,
    ! [B2: complex,A: complex] :
      ( ( B2 != zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2)) ) ) ).

% nonzero_norm_divide
tff(fact_6191_power__eq__imp__eq__norm,axiom,
    ! [W: real,Nb: nat,Zb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_real_real(real_V7735802525324610683m_real,W) = aa_real_real(real_V7735802525324610683m_real,Zb) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6192_power__eq__imp__eq__norm,axiom,
    ! [W: complex,Nb: nat,Zb: complex] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = aa_nat_complex(power_power_complex(Zb),Nb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_complex_real(real_V1022390504157884413omplex,W) = aa_complex_real(real_V1022390504157884413omplex,Zb) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_6193_norm__mult__less,axiom,
    ! [Xb: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_6194_norm__mult__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),S)) ) ) ).

% norm_mult_less
tff(fact_6195_norm__mult__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_mult_ineq
tff(fact_6196_norm__mult__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_mult_ineq
tff(fact_6197_norm__triangle__lt,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_6198_norm__triangle__lt,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E) ) ).

% norm_triangle_lt
tff(fact_6199_norm__add__less,axiom,
    ! [Xb: real,R2: real,Y: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_6200_norm__add__less,axiom,
    ! [Xb: complex,R2: real,Y: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Y)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_add_less
tff(fact_6201_norm__power__ineq,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V7735802525324610683m_real,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6202_norm__power__ineq,axiom,
    ! [Xb: complex,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_nat_complex(power_power_complex(Xb),Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),Nb)) ).

% norm_power_ineq
tff(fact_6203_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B2: real,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_6204_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B2: complex,S: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,A)),R2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),S)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),S)) ) ) ).

% norm_triangle_mono
tff(fact_6205_norm__triangle__ineq,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))) ).

% norm_triangle_ineq
tff(fact_6206_norm__triangle__ineq,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))) ).

% norm_triangle_ineq
tff(fact_6207_norm__triangle__le,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_6208_norm__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(Xb),Y))),E) ) ).

% norm_triangle_le
tff(fact_6209_norm__add__leD,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),C)) ) ).

% norm_add_leD
tff(fact_6210_norm__add__leD,axiom,
    ! [A: complex,B2: complex,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))),C)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),C)) ) ).

% norm_add_leD
tff(fact_6211_norm__diff__triangle__less,axiom,
    ! [Xb: real,Y: real,E1: real,Zb: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Zb))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_6212_norm__diff__triangle__less,axiom,
    ! [Xb: complex,Y: complex,E1: real,Zb: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Zb))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_less
tff(fact_6213_norm__triangle__sub,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Y)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)))) ).

% norm_triangle_sub
tff(fact_6214_norm__triangle__sub,axiom,
    ! [Xb: complex,Y: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Y)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y)))) ).

% norm_triangle_sub
tff(fact_6215_norm__triangle__ineq4,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2))) ).

% norm_triangle_ineq4
tff(fact_6216_norm__triangle__ineq4,axiom,
    ! [A: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2))) ).

% norm_triangle_ineq4
tff(fact_6217_norm__diff__triangle__le,axiom,
    ! [Xb: real,Y: real,E1: real,Zb: real,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Zb))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_6218_norm__diff__triangle__le,axiom,
    ! [Xb: complex,Y: complex,E1: real,Zb: complex,E22: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E1)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Y),Zb))),E22)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Zb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,E1),E22)) ) ) ).

% norm_diff_triangle_le
tff(fact_6219_norm__triangle__le__diff,axiom,
    ! [Xb: real,Y: real,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_6220_norm__triangle__le__diff,axiom,
    ! [Xb: complex,Y: complex,E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))),E)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y))),E) ) ).

% norm_triangle_le_diff
tff(fact_6221_norm__diff__ineq,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))) ).

% norm_diff_ineq
tff(fact_6222_norm__diff__ineq,axiom,
    ! [A: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(A),B2))) ).

% norm_diff_ineq
tff(fact_6223_norm__triangle__ineq2,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% norm_triangle_ineq2
tff(fact_6224_norm__triangle__ineq2,axiom,
    ! [A: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2))) ).

% norm_triangle_ineq2
tff(fact_6225_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_6226_Suc__times__binomial__add,axiom,
    ! [A: nat,B2: 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),B2))),aa_nat_nat(suc,A))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,B2)),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),B2))),A)) ).

% Suc_times_binomial_add
tff(fact_6227_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_6228_choose__mult,axiom,
    ! [K: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Mb),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,Mb),K))) ) ) ) ).

% choose_mult
tff(fact_6229_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_6230_power__eq__1__iff,axiom,
    ! [W: real,Nb: nat] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Nb) = one_one_real )
     => ( ( aa_real_real(real_V7735802525324610683m_real,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6231_power__eq__1__iff,axiom,
    ! [W: complex,Nb: nat] :
      ( ( aa_nat_complex(power_power_complex(W),Nb) = one_one_complex )
     => ( ( aa_complex_real(real_V1022390504157884413omplex,W) = one_one_real )
        | ( Nb = zero_zero_nat ) ) ) ).

% power_eq_1_iff
tff(fact_6232_norm__diff__triangle__ineq,axiom,
    ! [A: real,B2: real,C: real,D: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6233_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B2: complex,C: complex,D: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),B2)),aa_complex_complex(plus_plus_complex(C),D)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),C))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(B2),D)))) ).

% norm_diff_triangle_ineq
tff(fact_6234_norm__triangle__ineq3,axiom,
    ! [A: real,B2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V7735802525324610683m_real,A)),aa_real_real(real_V7735802525324610683m_real,B2)))),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))) ).

% norm_triangle_ineq3
tff(fact_6235_norm__triangle__ineq3,axiom,
    ! [A: complex,B2: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(real_V1022390504157884413omplex,A)),aa_complex_real(real_V1022390504157884413omplex,B2)))),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(A),B2))) ).

% norm_triangle_ineq3
tff(fact_6236_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_6237_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6238_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_6239_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_6240_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_6241_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_6242_square__norm__one,axiom,
    ! [Xb: real] :
      ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_real )
     => ( aa_real_real(real_V7735802525324610683m_real,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_6243_square__norm__one,axiom,
    ! [Xb: complex] :
      ( ( aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = one_one_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,Xb) = one_one_real ) ) ).

% square_norm_one
tff(fact_6244_norm__power__diff,axiom,
    ! [Zb: real,W: real,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,Zb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Zb),W)))) ) ) ).

% norm_power_diff
tff(fact_6245_norm__power__diff,axiom,
    ! [Zb: complex,W: complex,Mb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,W)),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(Zb),Mb)),aa_nat_complex(power_power_complex(W),Mb)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Zb),W)))) ) ) ).

% norm_power_diff
tff(fact_6246_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_6247_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_6248_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_6249_cos__pi__eq__zero,axiom,
    ! [Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,pi),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb))))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_pi_eq_zero
tff(fact_6250_round__altdef,axiom,
    ! [Xb: real] :
      aa_real_int(archim8280529875227126926d_real,Xb) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_int(archim7802044766580827645g_real,Xb),aa_real_int(archim6058952711729229775r_real,Xb)) ).

% round_altdef
tff(fact_6251_round__altdef,axiom,
    ! [Xb: rat] :
      aa_rat_int(archim7778729529865785530nd_rat,Xb) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_int(archim2889992004027027881ng_rat,Xb),aa_rat_int(archim3151403230148437115or_rat,Xb)) ).

% round_altdef
tff(fact_6252_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_nat_real(semiri5074537144036343181t_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),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_6253_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_6254_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cot_real,Xb)),zero_zero_real) ) ) ).

% cot_less_zero
tff(fact_6255_powr__one__eq__one,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,one_one_real),A) = one_one_real ).

% powr_one_eq_one
tff(fact_6256_cos__zero,axiom,
    aa_complex_complex(cos_complex,zero_zero_complex) = one_one_complex ).

% cos_zero
tff(fact_6257_cos__zero,axiom,
    aa_real_real(cos_real,zero_zero_real) = one_one_real ).

% cos_zero
tff(fact_6258_powr__zero__eq__one,axiom,
    ! [Xb: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),zero_zero_real) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% powr_zero_eq_one
tff(fact_6259_powr__gt__zero,axiom,
    ! [Xb: 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,Xb),A))
    <=> ( Xb != zero_zero_real ) ) ).

% powr_gt_zero
tff(fact_6260_powr__nonneg__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real)
    <=> ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
tff(fact_6261_powr__less__cancel__iff,axiom,
    ! [Xb: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_6262_sin__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(sin_real,Xb) ).

% sin_pi_minus
tff(fact_6263_powr__eq__one__iff,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),A)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = one_one_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% powr_eq_one_iff
tff(fact_6264_powr__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb ) ) ).

% powr_one
tff(fact_6265_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),one_one_real) = Xb )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_6266_powr__le__cancel__iff,axiom,
    ! [Xb: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_6267_numeral__powr__numeral__real,axiom,
    ! [Mb: num,Nb: num] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,Mb)),aa_num_nat(numeral_numeral_nat,Nb)) ).

% numeral_powr_numeral_real
tff(fact_6268_cos__pi,axiom,
    aa_real_real(cos_real,pi) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_pi
tff(fact_6269_cos__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi2
tff(fact_6270_cos__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_periodic_pi
tff(fact_6271_sin__periodic__pi2,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi2
tff(fact_6272_sin__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_periodic_pi
tff(fact_6273_cos__pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_pi_minus
tff(fact_6274_cos__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(cos_real,Xb)) ).

% cos_minus_pi
tff(fact_6275_sin__minus__pi,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),pi)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_minus_pi
tff(fact_6276_sin__cos__squared__add3,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Xb))),aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,Xb)),aa_complex_complex(sin_complex,Xb))) = one_one_complex ).

% sin_cos_squared_add3
tff(fact_6277_sin__cos__squared__add3,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Xb))) = one_one_real ).

% sin_cos_squared_add3
tff(fact_6278_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_6279_powr__log__cancel,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_6280_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_6281_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_6282_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_6283_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_6284_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_num_real(numeral_numeral_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,Nb)) ) ) ).

% powr_numeral
tff(fact_6285_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_6286_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_6287_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_6288_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_6289_cos__periodic,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cos_real,Xb) ).

% cos_periodic
tff(fact_6290_sin__periodic,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(sin_real,Xb) ).

% sin_periodic
tff(fact_6291_cos__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(cos_real,Xb) ).

% cos_2pi_minus
tff(fact_6292_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_6293_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_6294_cot__periodic,axiom,
    ! [Xb: real] : aa_real_real(cot_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(cot_real,Xb) ).

% cot_periodic
tff(fact_6295_sin__cos__squared__add2,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add2
tff(fact_6296_sin__cos__squared__add2,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add2
tff(fact_6297_sin__cos__squared__add,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% sin_cos_squared_add
tff(fact_6298_sin__cos__squared__add,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% sin_cos_squared_add
tff(fact_6299_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_6300_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_6301_sin__2pi__minus,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) ).

% sin_2pi_minus
tff(fact_6302_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_6303_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_6304_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_6305_square__powr__half,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(abs_abs_real,Xb) ).

% square_powr_half
tff(fact_6306_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_6307_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_6308_sin__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_add
tff(fact_6309_sin__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(sin_real,Y))) ).

% sin_diff
tff(fact_6310_cot__def,axiom,
    ! [X4: complex] : aa_complex_complex(cot_complex,X4) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,X4)),aa_complex_complex(sin_complex,X4)) ).

% cot_def
tff(fact_6311_cot__def,axiom,
    ! [X4: real] : aa_real_real(cot_real,X4) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,X4)),aa_real_real(sin_real,X4)) ).

% cot_def
tff(fact_6312_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R3: real,A4: real] :
      ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(cos_real,A4)) )
      & ( Y = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R3),aa_real_real(sin_real,A4)) ) ) ).

% polar_Ex
tff(fact_6313_cos__one__sin__zero,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) = one_one_complex )
     => ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex ) ) ).

% cos_one_sin_zero
tff(fact_6314_cos__one__sin__zero,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
     => ( aa_real_real(sin_real,Xb) = zero_zero_real ) ) ).

% cos_one_sin_zero
tff(fact_6315_powr__powr,axiom,
    ! [Xb: real,A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),B2) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) ).

% powr_powr
tff(fact_6316_cos__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_add
tff(fact_6317_cos__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))) ).

% cos_diff
tff(fact_6318_sin__zero__norm__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(real_V7735802525324610683m_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6319_sin__zero__norm__cos__one,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(sin_complex,Xb) = zero_zero_complex )
     => ( aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(cos_complex,Xb)) = one_one_real ) ) ).

% sin_zero_norm_cos_one
tff(fact_6320_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb)) = one_one_real ) ) ).

% sin_zero_abs_cos_one
tff(fact_6321_sin__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,Xb))),aa_complex_complex(cos_complex,Xb)) ).

% sin_double
tff(fact_6322_sin__double,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,Xb))),aa_real_real(cos_real,Xb)) ).

% sin_double
tff(fact_6323_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y3: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Y3)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y3),pi)
      & ( aa_real_real(sin_real,Y3) = aa_real_real(sin_real,Xb) )
      & ( aa_real_real(cos_real,Y3) = aa_real_real(cos_real,Xb) ) ) ).

% sincos_principal_value
tff(fact_6324_powr__less__mono2__neg,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_6325_powr__non__neg,axiom,
    ! [A: real,Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb)),zero_zero_real) ).

% powr_non_neg
tff(fact_6326_powr__mono2,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mono2
tff(fact_6327_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ).

% powr_ge_pzero
tff(fact_6328_powr__less__mono,axiom,
    ! [A: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2)) ) ) ).

% powr_less_mono
tff(fact_6329_powr__less__cancel,axiom,
    ! [Xb: real,A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% powr_less_cancel
tff(fact_6330_powr__mono,axiom,
    ! [A: real,B2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2)) ) ) ).

% powr_mono
tff(fact_6331_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_6332_sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),one_one_real) ).

% sin_le_one
tff(fact_6333_cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),one_one_real) ).

% cos_le_one
tff(fact_6334_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),aa_real_real(abs_abs_real,Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_6335_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))))),one_one_real) ).

% sin_cos_le1
tff(fact_6336_frac__ge__0,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb)) ).

% frac_ge_0
tff(fact_6337_frac__ge__0,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xb)) ).

% frac_ge_0
tff(fact_6338_sin__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_6339_sin__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% sin_squared_eq
tff(fact_6340_cos__squared__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_6341_cos__squared__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_squared_eq
tff(fact_6342_frac__lt__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(archim2898591450579166408c_real,Xb)),one_one_real) ).

% frac_lt_1
tff(fact_6343_frac__lt__1,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(archimedean_frac_rat,Xb)),one_one_rat) ).

% frac_lt_1
tff(fact_6344_frac__1__eq,axiom,
    ! [Xb: real] : aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) = aa_real_real(archim2898591450579166408c_real,Xb) ).

% frac_1_eq
tff(fact_6345_frac__1__eq,axiom,
    ! [Xb: rat] : aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),one_one_rat)) = aa_rat_rat(archimedean_frac_rat,Xb) ).

% frac_1_eq
tff(fact_6346_powr__mono2_H,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ) ).

% powr_mono2'
tff(fact_6347_powr__less__mono2,axiom,
    ! [A: real,Xb: 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),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_less_mono2
tff(fact_6348_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) ) ) ).

% gr_one_powr
tff(fact_6349_powr__inj,axiom,
    ! [A: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_6350_powr__le1,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),one_one_real) ) ) ) ).

% powr_le1
tff(fact_6351_powr__mono__both,axiom,
    ! [A: real,B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_6352_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_6353_powr__divide,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_divide
tff(fact_6354_powr__mult,axiom,
    ! [Xb: real,Y: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)),A) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Y),A)) ) ) ) ).

% powr_mult
tff(fact_6355_divide__powr__uminus,axiom,
    ! [A: real,B2: real,C: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),C)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,C))) ).

% divide_powr_uminus
tff(fact_6356_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_6357_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,Xb)),aa_real_real(sin_real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_6358_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_6359_log__base__powr,axiom,
    ! [A: real,B2: real,Xb: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),B2)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)),B2) ) ) ).

% log_base_powr
tff(fact_6360_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_powr
tff(fact_6361_log__powr,axiom,
    ! [Xb: real,B2: real,Y: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ).

% log_powr
tff(fact_6362_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(sin_real,Xb)) ).

% sin_ge_minus_one
tff(fact_6363_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_6364_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_6365_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( ( aa_real_real(cos_real,Xb) = aa_real_real(cos_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_6366_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),aa_real_real(cos_real,Xb)) ).

% cos_ge_minus_one
tff(fact_6367_abs__sin__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(sin_real,Xb))),one_one_real) ).

% abs_sin_le_one
tff(fact_6368_abs__cos__le__one,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(cos_real,Xb))),one_one_real) ).

% abs_cos_le_one
tff(fact_6369_cos__diff__cos,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Zb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(Zb),W)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_6370_cos__diff__cos,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Zb),W)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_diff_cos
tff(fact_6371_sin__diff__sin,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Zb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_6372_sin__diff__sin,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_diff_sin
tff(fact_6373_sin__plus__sin,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Zb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_6374_sin__plus__sin,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_plus_sin
tff(fact_6375_cos__times__sin,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(sin_complex,Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Zb))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_6376_cos__times__sin,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(sin_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_sin
tff(fact_6377_sin__times__cos,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(cos_complex,Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(W),Zb))),aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(W),Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_6378_sin__times__cos,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(cos_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb))),aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_cos
tff(fact_6379_sin__times__sin,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sin_complex,W)),aa_complex_complex(sin_complex,Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Zb))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_6380_sin__times__sin,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sin_real,W)),aa_real_real(sin_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sin_times_sin
tff(fact_6381_cos__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_6382_cos__double,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cos_double
tff(fact_6383_powr__add,axiom,
    ! [Xb: real,A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),B2)) ).

% powr_add
tff(fact_6384_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Z1),Z22)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z1)),aa_real_real(aa_rea7340605660300375871l_real(powr_real,W),Z22)) ).

% powr_diff
tff(fact_6385_cos__double__sin,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(sin_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_6386_cos__double__sin,axiom,
    ! [W: 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))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sin_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% cos_double_sin
tff(fact_6387_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_nat_real(semiri5074537144036343181t_real,Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb) ) ) ).

% powr_realpow
tff(fact_6388_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_6389_less__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_6390_log__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)) ) ) ) ).

% log_less_iff
tff(fact_6391_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_6392_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_6393_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_6394_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),pi)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_6395_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),pi)
       => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
         => ( Xb = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
tff(fact_6396_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),pi)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
tff(fact_6397_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_6398_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I2: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_6399_frac__def,axiom,
    ! [Xb: real] : aa_real_real(archim2898591450579166408c_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,Xb))) ).

% frac_def
tff(fact_6400_frac__def,axiom,
    ! [Xb: rat] : aa_rat_rat(archimedean_frac_rat,Xb) = aa_rat_rat(minus_minus_rat(Xb),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,Xb))) ).

% frac_def
tff(fact_6401_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),pi)
            & ( Xb = aa_real_real(cos_real,T) )
            & ( Y = aa_real_real(sin_real,T) ) ) ) ) ).

% sincos_total_pi
tff(fact_6402_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sin_real,Xb))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% sin_cos_sqrt
tff(fact_6403_sin__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% sin_expansion_lemma
tff(fact_6404_powr__minus__divide,axiom,
    ! [Xb: real,A: real] : aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,A)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)) ).

% powr_minus_divide
tff(fact_6405_cos__expansion__lemma,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Mb))),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) = aa_real_real(uminus_uminus_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% cos_expansion_lemma
tff(fact_6406_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb) ) ) ).

% powr_neg_one
tff(fact_6407_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_6408_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),Y)) ) ) ).

% powr_mult_base
tff(fact_6409_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_6410_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_6411_log__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y)
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)) ) ) ) ).

% log_le_iff
tff(fact_6412_le__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_6413_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_6414_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_6415_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_6416_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,Y)),aa_real_real(cos_real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_6417_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_6418_frac__eq,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = Xb )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% frac_eq
tff(fact_6419_frac__eq,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xb) = Xb )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ) ).

% frac_eq
tff(fact_6420_frac__add,axiom,
    ! [Xb: real,Y: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real)) ).

% frac_add
tff(fact_6421_frac__add,axiom,
    ! [Xb: rat,Y: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y)),aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat)) ).

% frac_add
tff(fact_6422_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
              & ( Xb = aa_real_real(cos_real,T) )
              & ( Y = aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_6423_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
          & ( Xb = aa_real_real(cos_real,T) )
          & ( Y = aa_real_real(sin_real,T) ) ) ) ).

% sincos_total_2pi_le
tff(fact_6424_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( ( Xb = aa_real_real(cos_real,T) )
               => ( Y != aa_real_real(sin_real,T) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_6425_ln__powr__bound,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(ln_ln_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A)),A)) ) ) ).

% ln_powr_bound
tff(fact_6426_ln__powr__bound2,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(ln_ln_real,Xb)),A)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,A),A)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_6427_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_6428_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_6429_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_6430_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_6431_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_6432_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_6433_powr__def,axiom,
    ! [Xb: real,A: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),A) = $ite(Xb = zero_zero_real,zero_zero_real,aa_real_real(exp_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(ln_ln_real,Xb)))) ).

% powr_def
tff(fact_6434_cos__plus__cos,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Zb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one)))))),aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(W),Zb)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_6435_cos__plus__cos,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% cos_plus_cos
tff(fact_6436_cos__times__cos,axiom,
    ! [W: complex,Zb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,W)),aa_complex_complex(cos_complex,Zb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(W),Zb))),aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(W),Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_6437_cos__times__cos,axiom,
    ! [W: real,Zb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,W)),aa_real_real(cos_real,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,W),Zb))),aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,W),Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cos_times_cos
tff(fact_6438_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sin_real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_6439_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_lt_zero
tff(fact_6440_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real) ) ) ).

% cos_double_less_one
tff(fact_6441_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_6442_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_6443_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_6444_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_6445_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( ( aa_real_real(sin_real,Xb) = aa_real_real(sin_real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_6446_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
     => ( ( B2 != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)),Y) = aa_real_real(aa_rea7340605660300375871l_real(log,B2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_real_real(uminus_uminus_real,Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_6447_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_6448_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_6449_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_6450_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ? [X: int] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi) ) ).

% cos_one_2pi_int
tff(fact_6451_cos__double__cos,axiom,
    ! [W: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),W)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_complex) ).

% cos_double_cos
tff(fact_6452_cos__double__cos,axiom,
    ! [W: 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))),W)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,W)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),one_one_real) ).

% cos_double_cos
tff(fact_6453_cos__treble__cos,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),Xb)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(cos_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit1,one))),aa_complex_complex(cos_complex,Xb))) ).

% cos_treble_cos
tff(fact_6454_cos__treble__cos,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),Xb)) = 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_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cos_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,one))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit1,one))),aa_real_real(cos_real,Xb))) ).

% cos_treble_cos
tff(fact_6455_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = aa_real_real(sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_6456_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(uminus_uminus_real,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,Xb),aa_num_nat(numeral_numeral_nat,Nb))) ) ) ).

% powr_neg_numeral
tff(fact_6457_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,pi),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_le_zero
tff(fact_6458_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),zero_zero_real) ) ) ).

% sin_less_zero
tff(fact_6459_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Y)),aa_real_real(sin_real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_6460_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sin_real,Xb)),aa_real_real(sin_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_6461_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_6462_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_6463_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cos_real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_6464_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = one_one_real )
    <=> ( ? [X: nat] : Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)
        | ? [X: nat] : Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,X)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_6465_floor__add,axiom,
    ! [Xb: real,Y: real] :
      aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(archim2898591450579166408c_real,Xb)),aa_real_real(archim2898591450579166408c_real,Y))),one_one_real),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y))),one_one_int)) ).

% floor_add
tff(fact_6466_floor__add,axiom,
    ! [Xb: rat,Y: rat] :
      aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)) = $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(archimedean_frac_rat,Xb)),aa_rat_rat(archimedean_frac_rat,Y))),one_one_rat),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y))),one_one_int)) ).

% floor_add
tff(fact_6467_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_6468_sin__arctan,axiom,
    ! [Xb: real] : aa_real_real(sin_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% sin_arctan
tff(fact_6469_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
       => ( ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) = K )
        <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,K))),Xb)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,B2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int)))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_6470_cos__arctan,axiom,
    ! [Xb: real] : aa_real_real(cos_real,aa_real_real(arctan,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% cos_arctan
tff(fact_6471_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cot_real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_6472_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ? [I2: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),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_6473_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
    <=> ? [I2: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),I2)
          & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I2)),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_6474_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),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_6475_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,Xb) = zero_zero_real )
    <=> ( ? [N4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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)
            & ( Xb = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N4)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_6476_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( aa_real_real(cos_real,Xb) = zero_zero_real )
       => ? [N2: nat] :
            ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N2)
            & ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,N2)),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_6477_tan__double,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) != zero_zero_complex )
       => ( aa_complex_complex(tan_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(tan_complex,Xb))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_6478_tan__double,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) != zero_zero_real )
       => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(tan_real,Xb))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% tan_double
tff(fact_6479_sin__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(tan_real,Xb)),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% sin_tan
tff(fact_6480_cos__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
     => ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ) ) ).

% cos_tan
tff(fact_6481_arcosh__def,axiom,
    ! [Xb: real] : aa_real_real(arcosh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arcosh_def
tff(fact_6482_complex__unimodular__polar,axiom,
    ! [Zb: complex] :
      ( ( aa_complex_real(real_V1022390504157884413omplex,Zb) = one_one_real )
     => ~ ! [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,T),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))
             => ( Zb != aa_real_complex(complex2(aa_real_real(cos_real,T)),aa_real_real(sin_real,T)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_6483_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% cos_arcsin
tff(fact_6484_of__real__1,axiom,
    aa_real_real(real_V1803761363581548252l_real,one_one_real) = one_one_real ).

% of_real_1
tff(fact_6485_of__real__1,axiom,
    aa_real_complex(real_V4546457046886955230omplex,one_one_real) = one_one_complex ).

% of_real_1
tff(fact_6486_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(real_V1803761363581548252l_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6487_of__real__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_complex(real_V4546457046886955230omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_real ) ) ).

% of_real_eq_1_iff
tff(fact_6488_of__real__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_mult
tff(fact_6489_of__real__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_mult
tff(fact_6490_of__real__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_num_real(numeral_numeral_real,W)) = aa_num_real(numeral_numeral_real,W) ).

% of_real_numeral
tff(fact_6491_of__real__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_num_real(numeral_numeral_real,W)) = numera6690914467698888265omplex(W) ).

% of_real_numeral
tff(fact_6492_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_divide
tff(fact_6493_of__real__divide,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_divide
tff(fact_6494_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(real_V1803761363581548252l_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(real_V1803761363581548252l_real,Xb)),Nb) ).

% of_real_power
tff(fact_6495_of__real__power,axiom,
    ! [Xb: real,Nb: nat] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = aa_nat_complex(power_power_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),Nb) ).

% of_real_power
tff(fact_6496_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_add
tff(fact_6497_of__real__add,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_add
tff(fact_6498_of__real__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ).

% of_real_diff
tff(fact_6499_of__real__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ).

% of_real_diff
tff(fact_6500_tan__periodic__pi,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),pi)) = aa_real_real(tan_real,Xb) ).

% tan_periodic_pi
tff(fact_6501_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_real(real_V1803761363581548252l_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) ).

% of_real_neg_numeral
tff(fact_6502_of__real__neg__numeral,axiom,
    ! [W: num] : aa_real_complex(real_V4546457046886955230omplex,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) ).

% of_real_neg_numeral
tff(fact_6503_cos__of__real__pi,axiom,
    aa_real_real(cos_real,aa_real_real(real_V1803761363581548252l_real,pi)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% cos_of_real_pi
tff(fact_6504_cos__of__real__pi,axiom,
    aa_complex_complex(cos_complex,aa_real_complex(real_V4546457046886955230omplex,pi)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% cos_of_real_pi
tff(fact_6505_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_6506_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_n
tff(fact_6507_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_6508_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_nat
tff(fact_6509_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,I)),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic_int
tff(fact_6510_norm__cos__sin,axiom,
    ! [Ta: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(aa_real_real(cos_real,Ta)),aa_real_real(sin_real,Ta))) = one_one_real ).

% norm_cos_sin
tff(fact_6511_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),one_one_real)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_6512_norm__of__real__add1,axiom,
    ! [Xb: real] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),one_one_complex)) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),one_one_real)) ).

% norm_of_real_add1
tff(fact_6513_norm__of__real__addn,axiom,
    ! [Xb: real,B2: num] : aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_num_real(numeral_numeral_real,B2))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,B2))) ).

% norm_of_real_addn
tff(fact_6514_norm__of__real__addn,axiom,
    ! [Xb: real,B2: num] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,Xb)),numera6690914467698888265omplex(B2))) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_num_real(numeral_numeral_real,B2))) ).

% norm_of_real_addn
tff(fact_6515_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_6516_tan__periodic,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) = aa_real_real(tan_real,Xb) ).

% tan_periodic
tff(fact_6517_cos__of__real__pi__half,axiom,
    aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = zero_zero_real ).

% cos_of_real_pi_half
tff(fact_6518_cos__of__real__pi__half,axiom,
    aa_complex_complex(cos_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = zero_zero_complex ).

% cos_of_real_pi_half
tff(fact_6519_sin__of__real__pi__half,axiom,
    aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) = one_one_real ).

% sin_of_real_pi_half
tff(fact_6520_sin__of__real__pi__half,axiom,
    aa_complex_complex(sin_complex,aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) = one_one_complex ).

% sin_of_real_pi_half
tff(fact_6521_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_6522_complex__of__real__mult__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Xb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),Y)) ).

% complex_of_real_mult_Complex
tff(fact_6523_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),R2)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),R2)) ).

% Complex_mult_complex_of_real
tff(fact_6524_complex__of__real__add__Complex,axiom,
    ! [R2: real,Xb: real,Y: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(complex2(Xb),Y)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,R2),Xb)),Y) ).

% complex_of_real_add_Complex
tff(fact_6525_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Y: real,R2: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(Xb),Y)),aa_real_complex(real_V4546457046886955230omplex,R2)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),R2)),Y) ).

% Complex_add_complex_of_real
tff(fact_6526_complex__diff,axiom,
    ! [A: real,B2: real,C: real,D: real] : aa_complex_complex(minus_minus_complex(aa_real_complex(complex2(A),B2)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),D)) ).

% complex_diff
tff(fact_6527_Complex__eq__1,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_complex(complex2(A),B2) = one_one_complex )
    <=> ( ( A = one_one_real )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_1
tff(fact_6528_one__complex_Ocode,axiom,
    one_one_complex = aa_real_complex(complex2(one_one_real),zero_zero_real) ).

% one_complex.code
tff(fact_6529_Complex__eq__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( aa_real_complex(complex2(A),B2) = numera6690914467698888265omplex(W) )
    <=> ( ( A = aa_num_real(numeral_numeral_real,W) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_numeral
tff(fact_6530_complex__add,axiom,
    ! [A: real,B2: real,C: real,D: real] : aa_complex_complex(plus_plus_complex(aa_real_complex(complex2(A),B2)),aa_real_complex(complex2(C),D)) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),C)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,B2),D)) ).

% complex_add
tff(fact_6531_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,Xb)),aa_real_real(real_V1803761363581548252l_real,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_6532_nonzero__of__real__divide,axiom,
    ! [Y: real,Xb: real] :
      ( ( Y != zero_zero_real )
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,Xb)),aa_real_complex(real_V4546457046886955230omplex,Y)) ) ) ).

% nonzero_of_real_divide
tff(fact_6533_Complex__eq__neg__1,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_complex(complex2(A),B2) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,one_one_real) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
tff(fact_6534_Complex__eq__neg__numeral,axiom,
    ! [A: real,B2: real,W: num] :
      ( ( aa_real_complex(complex2(A),B2) = aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W)) )
    <=> ( ( A = aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W)) )
        & ( B2 = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_6535_complex__mult,axiom,
    ! [A: real,B2: real,C: real,D: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B2)),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,B2),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,B2),C))) ).

% complex_mult
tff(fact_6536_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_6537_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arcsin,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(arcsin,Xb)) ) ) ) ).

% arcsin_minus
tff(fact_6538_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_6539_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( ( aa_real_real(arcsin,Xb) = aa_real_real(arcsin,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_6540_norm__less__p1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,Xb)),aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(real_V1803761363581548252l_real,aa_real_real(real_V7735802525324610683m_real,Xb))),one_one_real))) ).

% norm_less_p1
tff(fact_6541_norm__less__p1,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Xb))),one_one_complex))) ).

% norm_less_p1
tff(fact_6542_tan__def,axiom,
    ! [X4: complex] : aa_complex_complex(tan_complex,X4) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,X4)),aa_complex_complex(cos_complex,X4)) ).

% tan_def
tff(fact_6543_tan__def,axiom,
    ! [X4: real] : aa_real_real(tan_real,X4) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,X4)),aa_real_real(cos_real,X4)) ).

% tan_def
tff(fact_6544_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_6545_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arcsin,Xb)),aa_real_real(arcsin,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_6546_norm__of__real__diff,axiom,
    ! [B2: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(real_V1803761363581548252l_real,B2)),aa_real_real(real_V1803761363581548252l_real,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% norm_of_real_diff
tff(fact_6547_norm__of__real__diff,axiom,
    ! [B2: real,A: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_real_complex(real_V4546457046886955230omplex,B2)),aa_real_complex(real_V4546457046886955230omplex,A)))),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))) ).

% norm_of_real_diff
tff(fact_6548_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_6549_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_6550_cos__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% cos_int_times_real
tff(fact_6551_cos__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% cos_int_times_real
tff(fact_6552_sin__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),aa_real_real(real_V1803761363581548252l_real,Xb))) = aa_real_real(real_V1803761363581548252l_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% sin_int_times_real
tff(fact_6553_sin__int__times__real,axiom,
    ! [Mb: int,Xb: real] : aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(aa_int_complex(ring_17405671764205052669omplex,Mb)),aa_real_complex(real_V4546457046886955230omplex,Xb))) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,Mb)),Xb))) ).

% sin_int_times_real
tff(fact_6554_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arcsin,Xb)) != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
tff(fact_6555_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_6556_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_6557_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_6558_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_6559_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Y)),aa_real_real(tan_real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_6560_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_6561_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_6562_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_6563_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_6564_complex__norm,axiom,
    ! [Xb: real,Y: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(complex2(Xb),Y)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_norm
tff(fact_6565_add__tan__eq,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% add_tan_eq
tff(fact_6566_add__tan__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% add_tan_eq
tff(fact_6567_tan__cot_H,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(cot_real,Xb) ).

% tan_cot'
tff(fact_6568_sin__cos__eq,axiom,
    ! [Xb: real] : aa_real_real(sin_real,Xb) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).

% sin_cos_eq
tff(fact_6569_sin__cos__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(sin_complex,Xb) = aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Xb)) ).

% sin_cos_eq
tff(fact_6570_cos__sin__eq,axiom,
    ! [Xb: real] : aa_real_real(cos_real,Xb) = aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) ).

% cos_sin_eq
tff(fact_6571_cos__sin__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(cos_complex,Xb) = aa_complex_complex(sin_complex,aa_complex_complex(minus_minus_complex(aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Xb)) ).

% cos_sin_eq
tff(fact_6572_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(tan_real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_6573_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_6574_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(tan_real,Xb)),zero_zero_real) ) ) ).

% tan_less_zero
tff(fact_6575_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_6576_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_6577_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(tan_real,Xb))),one_one_real) ) ).

% tan_bound_pi2
tff(fact_6578_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_6579_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_6580_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arctan,aa_real_real(tan_real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_6581_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( ( aa_real_real(tan_real,Xb) = Y )
         => ( aa_real_real(arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_6582_lemma__tan__add1,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y))),aa_complex_complex(times_times_complex(aa_complex_complex(cos_complex,Xb)),aa_complex_complex(cos_complex,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_6583_lemma__tan__add1,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cos_real,Xb)),aa_real_real(cos_real,Y))) ) ) ) ).

% lemma_tan_add1
tff(fact_6584_tan__diff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_6585_tan__diff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_diff
tff(fact_6586_tan__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cos_complex,Y) != zero_zero_complex )
       => ( ( aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) != zero_zero_complex )
         => ( aa_complex_complex(tan_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y))),aa_complex_complex(minus_minus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tan_complex,Xb)),aa_complex_complex(tan_complex,Y)))) ) ) ) ) ).

% tan_add
tff(fact_6587_tan__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cos_real,Y) != zero_zero_real )
       => ( ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) != zero_zero_real )
         => ( aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tan_real,Xb)),aa_real_real(tan_real,Y)))) ) ) ) ) ).

% tan_add
tff(fact_6588_minus__sin__cos__eq,axiom,
    ! [Xb: real] : aa_real_real(uminus_uminus_real,aa_real_real(sin_real,Xb)) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(real_V1803761363581548252l_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_6589_minus__sin__cos__eq,axiom,
    ! [Xb: complex] : aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sin_complex,Xb)) = aa_complex_complex(cos_complex,aa_complex_complex(plus_plus_complex(Xb),aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(real_V4546457046886955230omplex,pi)),numera6690914467698888265omplex(aa_num_num(bit0,one))))) ).

% minus_sin_cos_eq
tff(fact_6590_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ? [Z: 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)))))),Z)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,aa_num_num(bit0,one)))))
          & ( aa_real_real(tan_real,Z) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_6591_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_6592_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_6593_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_6594_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_6595_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
       => ( aa_real_real(arcsin,aa_real_real(sin_real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_6596_tan__half,axiom,
    ! [Xb: complex] : aa_complex_complex(tan_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sin_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),aa_complex_complex(plus_plus_complex(aa_complex_complex(cos_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb))),one_one_complex)) ).

% tan_half
tff(fact_6597_tan__half,axiom,
    ! [Xb: real] : aa_real_real(tan_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb))),one_one_real)) ).

% tan_half
tff(fact_6598_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(arcsin,Xb))
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sin_real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_6599_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Y)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arcsin,Xb)),Y)
            <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(sin_real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_6600_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_6601_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_6602_arsinh__def,axiom,
    ! [Xb: real] : aa_real_real(arsinh_real,Xb) = aa_real_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real)),aa_real_real(real_V1803761363581548252l_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))))) ).

% arsinh_def
tff(fact_6603_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_6604_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) = aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ).

% sin_arccos
tff(fact_6605_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_6606_fact__double,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one)))),Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_double
tff(fact_6607_fact__double,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))),Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_double
tff(fact_6608_fact__double,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_double
tff(fact_6609_modulo__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,K) = zero_zero_int )
        | ( Nb = zero_zero_nat ) ),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb)),
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb))))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb))))) ) ).

% modulo_int_unfold
tff(fact_6610_sgn__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(sgn_sgn_int,A)) = aa_int_int(sgn_sgn_int,A) ).

% sgn_sgn
tff(fact_6611_sgn__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% sgn_sgn
tff(fact_6612_sgn__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(sgn_sgn_complex,A)) = aa_complex_complex(sgn_sgn_complex,A) ).

% sgn_sgn
tff(fact_6613_sgn__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ).

% sgn_sgn
tff(fact_6614_sgn__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% sgn_sgn
tff(fact_6615_sgn__0,axiom,
    aa_complex_complex(sgn_sgn_complex,zero_zero_complex) = zero_zero_complex ).

% sgn_0
tff(fact_6616_sgn__0,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% sgn_0
tff(fact_6617_sgn__0,axiom,
    aa_real_real(sgn_sgn_real,zero_zero_real) = zero_zero_real ).

% sgn_0
tff(fact_6618_sgn__0,axiom,
    aa_rat_rat(sgn_sgn_rat,zero_zero_rat) = zero_zero_rat ).

% sgn_0
tff(fact_6619_sgn__0,axiom,
    aa_int_int(sgn_sgn_int,zero_zero_int) = zero_zero_int ).

% sgn_0
tff(fact_6620_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_6621_sgn__1,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_1
tff(fact_6622_sgn__1,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_1
tff(fact_6623_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_6624_sgn__1,axiom,
    aa_rat_rat(sgn_sgn_rat,one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_6625_sgn__one,axiom,
    aa_real_real(sgn_sgn_real,one_one_real) = one_one_real ).

% sgn_one
tff(fact_6626_sgn__one,axiom,
    aa_complex_complex(sgn_sgn_complex,one_one_complex) = one_one_complex ).

% sgn_one
tff(fact_6627_sgn__divide,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B2)) ).

% sgn_divide
tff(fact_6628_sgn__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B2)) ).

% sgn_divide
tff(fact_6629_sgn__divide,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B2)) ).

% sgn_divide
tff(fact_6630_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6631_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6632_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6633_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6634_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_6635_power__sgn,axiom,
    ! [A: code_integer,Nb: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,power_8256067586552552935nteger(A,Nb)) = power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A),Nb) ).

% power_sgn
tff(fact_6636_power__sgn,axiom,
    ! [A: rat,Nb: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(power_power_rat(A),Nb)) = aa_nat_rat(power_power_rat(aa_rat_rat(sgn_sgn_rat,A)),Nb) ).

% power_sgn
tff(fact_6637_power__sgn,axiom,
    ! [A: real,Nb: nat] : aa_real_real(sgn_sgn_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(sgn_sgn_real,A)),Nb) ).

% power_sgn
tff(fact_6638_power__sgn,axiom,
    ! [A: int,Nb: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(power_power_int(A),Nb)) = aa_nat_int(power_power_int(aa_int_int(sgn_sgn_int,A)),Nb) ).

% power_sgn
tff(fact_6639_sgn__greater,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_greater
tff(fact_6640_sgn__greater,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sgn_sgn_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_greater
tff(fact_6641_sgn__greater,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(sgn_sgn_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_greater
tff(fact_6642_sgn__greater,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(sgn_sgn_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_greater
tff(fact_6643_sgn__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_less
tff(fact_6644_sgn__less,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sgn_sgn_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_less
tff(fact_6645_sgn__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(sgn_sgn_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_less
tff(fact_6646_sgn__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(sgn_sgn_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_less
tff(fact_6647_divide__sgn,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_real_real(sgn_sgn_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(sgn_sgn_real,B2)) ).

% divide_sgn
tff(fact_6648_divide__sgn,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(divide_divide_rat(A),aa_rat_rat(sgn_sgn_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(sgn_sgn_rat,B2)) ).

% divide_sgn
tff(fact_6649_fact__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,zero_zero_nat) = one_one_complex ).

% fact_0
tff(fact_6650_fact__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,zero_zero_nat) = one_one_rat ).

% fact_0
tff(fact_6651_fact__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,zero_zero_nat) = one_one_int ).

% fact_0
tff(fact_6652_fact__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,zero_zero_nat) = one_one_nat ).

% fact_0
tff(fact_6653_fact__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,zero_zero_nat) = one_one_real ).

% fact_0
tff(fact_6654_fact__1,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,one_one_nat) = one_one_complex ).

% fact_1
tff(fact_6655_fact__1,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,one_one_nat) = one_one_rat ).

% fact_1
tff(fact_6656_fact__1,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,one_one_nat) = one_one_int ).

% fact_1
tff(fact_6657_fact__1,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,one_one_nat) = one_one_nat ).

% fact_1
tff(fact_6658_fact__1,axiom,
    aa_nat_real(semiri2265585572941072030t_real,one_one_nat) = one_one_real ).

% fact_1
tff(fact_6659_arccos__1,axiom,
    aa_real_real(arccos,one_one_real) = zero_zero_real ).

% arccos_1
tff(fact_6660_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_6661_sgn__pos,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_real(sgn_sgn_real,A) = one_one_real ) ) ).

% sgn_pos
tff(fact_6662_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_6663_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_6664_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_6665_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = one_one_real ) ) ).

% abs_sgn_eq_1
tff(fact_6666_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_6667_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_6668_fact__Suc__0,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,zero_zero_nat)) = one_one_complex ).

% fact_Suc_0
tff(fact_6669_fact__Suc__0,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,zero_zero_nat)) = one_one_rat ).

% fact_Suc_0
tff(fact_6670_fact__Suc__0,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% fact_Suc_0
tff(fact_6671_fact__Suc__0,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% fact_Suc_0
tff(fact_6672_fact__Suc__0,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,zero_zero_nat)) = one_one_real ).

% fact_Suc_0
tff(fact_6673_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,Nb))),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% fact_Suc
tff(fact_6674_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_Suc
tff(fact_6675_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,Nb))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_Suc
tff(fact_6676_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(suc,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_Suc
tff(fact_6677_fact__Suc,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_Suc
tff(fact_6678_sgn__mult__self__eq,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% sgn_mult_self_eq
tff(fact_6679_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_6680_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_6681_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_6682_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(abs_abs_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6683_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(abs_abs_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6684_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(abs_abs_rat,A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6685_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] : aa_int_int(sgn_sgn_int,aa_int_int(abs_abs_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6686_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,abs_abs_Code_integer(A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_6687_sgn__abs,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(sgn_sgn_complex,A)) = zero_n1201886186963655149omplex(A != zero_zero_complex) ).

% sgn_abs
tff(fact_6688_sgn__abs,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = zero_n3304061248610475627l_real(A != zero_zero_real) ).

% sgn_abs
tff(fact_6689_sgn__abs,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_abs
tff(fact_6690_sgn__abs,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = aa_o_int(zero_n2684676970156552555ol_int,A != zero_zero_int) ).

% sgn_abs
tff(fact_6691_sgn__abs,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_abs
tff(fact_6692_arccos__minus__1,axiom,
    aa_real_real(arccos,aa_real_real(uminus_uminus_real,one_one_real)) = pi ).

% arccos_minus_1
tff(fact_6693_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(aa_int_fun_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_6694_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(aa_int_fun_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_6695_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(aa_int_fun_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_6696_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(aa_int_fun_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_6697_sgn__neg,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% sgn_neg
tff(fact_6698_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_6699_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_6700_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ) ) ).

% sgn_neg
tff(fact_6701_fact__2,axiom,
    aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = numera6690914467698888265omplex(aa_num_num(bit0,one)) ).

% fact_2
tff(fact_6702_fact__2,axiom,
    aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_6703_fact__2,axiom,
    aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_6704_fact__2,axiom,
    aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_6705_fact__2,axiom,
    aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)) ).

% fact_2
tff(fact_6706_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_real_real(sgn_sgn_real,aa_nat_real(semiri5074537144036343181t_real,Nb)) = zero_n3304061248610475627l_real(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_6707_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_rat_rat(sgn_sgn_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)) = zero_n2052037380579107095ol_rat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_6708_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_int_int(sgn_sgn_int,aa_nat_int(semiri1314217659103216013at_int,Nb)) = aa_o_int(zero_n2684676970156552555ol_int,aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_6709_sgn__of__nat,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,semiri4939895301339042750nteger(Nb)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)) ).

% sgn_of_nat
tff(fact_6710_cos__arccos,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_6711_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_6712_complex__exp__exists,axiom,
    ! [Zb: complex] :
    ? [A4: complex,R3: real] : Zb = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(exp_complex,A4)) ).

% complex_exp_exists
tff(fact_6713_fact__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono_nat
tff(fact_6714_fact__ge__self,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_self
tff(fact_6715_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(sgn_sgn_complex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
tff(fact_6716_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
tff(fact_6717_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
tff(fact_6718_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
tff(fact_6719_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
tff(fact_6720_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
tff(fact_6721_sgn__0__0,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% sgn_0_0
tff(fact_6722_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% sgn_0_0
tff(fact_6723_sgn__0__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% sgn_0_0
tff(fact_6724_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,Xb)),aa_complex_complex(sgn_sgn_complex,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_6725_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),aa_real_real(sgn_sgn_real,Y)) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_6726_sgn__mult,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(sgn_sgn_complex,B2)) ).

% sgn_mult
tff(fact_6727_sgn__mult,axiom,
    ! [A: code_integer,B2: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2)) ).

% sgn_mult
tff(fact_6728_sgn__mult,axiom,
    ! [A: real,B2: real] : aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(sgn_sgn_real,B2)) ).

% sgn_mult
tff(fact_6729_sgn__mult,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(sgn_sgn_rat,B2)) ).

% sgn_mult
tff(fact_6730_sgn__mult,axiom,
    ! [A: int,B2: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B2)) ).

% sgn_mult
tff(fact_6731_same__sgn__sgn__add,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,plus_p5714425477246183910nteger(A,B2)) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) ) ) ).

% same_sgn_sgn_add
tff(fact_6732_same__sgn__sgn__add,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(sgn_sgn_real,A) ) ) ).

% same_sgn_sgn_add
tff(fact_6733_same__sgn__sgn__add,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B2) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(sgn_sgn_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(sgn_sgn_rat,A) ) ) ).

% same_sgn_sgn_add
tff(fact_6734_same__sgn__sgn__add,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(sgn_sgn_int,A) ) ) ).

% same_sgn_sgn_add
tff(fact_6735_fact__less__mono__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_6736_sgn__not__eq__imp,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) != aa_real_real(sgn_sgn_real,A) )
     => ( ( aa_real_real(sgn_sgn_real,A) != zero_zero_real )
       => ( ( aa_real_real(sgn_sgn_real,B2) != zero_zero_real )
         => ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,aa_real_real(sgn_sgn_real,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6737_sgn__not__eq__imp,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) != aa_int_int(sgn_sgn_int,A) )
     => ( ( aa_int_int(sgn_sgn_int,A) != zero_zero_int )
       => ( ( aa_int_int(sgn_sgn_int,B2) != zero_zero_int )
         => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,aa_int_int(sgn_sgn_int,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6738_sgn__not__eq__imp,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2) != aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) != zero_z3403309356797280102nteger )
       => ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2) != zero_z3403309356797280102nteger )
         => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6739_sgn__not__eq__imp,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B2) != aa_rat_rat(sgn_sgn_rat,A) )
     => ( ( aa_rat_rat(sgn_sgn_rat,A) != zero_zero_rat )
       => ( ( aa_rat_rat(sgn_sgn_rat,B2) != zero_zero_rat )
         => ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(sgn_sgn_rat,B2)) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_6740_sgn__minus__1,axiom,
    aa_real_real(sgn_sgn_real,aa_real_real(uminus_uminus_real,one_one_real)) = aa_real_real(uminus_uminus_real,one_one_real) ).

% sgn_minus_1
tff(fact_6741_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_6742_sgn__minus__1,axiom,
    aa_complex_complex(sgn_sgn_complex,aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% sgn_minus_1
tff(fact_6743_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_6744_sgn__minus__1,axiom,
    aa_rat_rat(sgn_sgn_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) = aa_rat_rat(uminus_uminus_rat,one_one_rat) ).

% sgn_minus_1
tff(fact_6745_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_zero
tff(fact_6746_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_zero
tff(fact_6747_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_zero
tff(fact_6748_fact__ge__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_zero
tff(fact_6749_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_6750_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: real] : aa_real_real(abs_abs_real,K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,K),aa_real_real(sgn_sgn_real,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_6751_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : aa_rat_rat(abs_abs_rat,K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),aa_rat_rat(sgn_sgn_rat,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_6752_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_6753_abs__mult__sgn,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(abs_abs_complex,A)),aa_complex_complex(sgn_sgn_complex,A)) = A ).

% abs_mult_sgn
tff(fact_6754_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_6755_abs__mult__sgn,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(abs_abs_real,A)),aa_real_real(sgn_sgn_real,A)) = A ).

% abs_mult_sgn
tff(fact_6756_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(sgn_sgn_rat,A)) = A ).

% abs_mult_sgn
tff(fact_6757_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_6758_sgn__mult__abs,axiom,
    ! [A: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(sgn_sgn_complex,A)),aa_complex_complex(abs_abs_complex,A)) = A ).

% sgn_mult_abs
tff(fact_6759_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_6760_sgn__mult__abs,axiom,
    ! [A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_real_real(abs_abs_real,A)) = A ).

% sgn_mult_abs
tff(fact_6761_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,A)),aa_rat_rat(abs_abs_rat,A)) = A ).

% sgn_mult_abs
tff(fact_6762_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_6763_mult__sgn__abs,axiom,
    ! [Xb: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xb)),abs_abs_Code_integer(Xb)) = Xb ).

% mult_sgn_abs
tff(fact_6764_mult__sgn__abs,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),aa_real_real(abs_abs_real,Xb)) = Xb ).

% mult_sgn_abs
tff(fact_6765_mult__sgn__abs,axiom,
    ! [Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(sgn_sgn_rat,Xb)),aa_rat_rat(abs_abs_rat,Xb)) = Xb ).

% mult_sgn_abs
tff(fact_6766_mult__sgn__abs,axiom,
    ! [Xb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Xb)),aa_int_int(abs_abs_int,Xb)) = Xb ).

% mult_sgn_abs
tff(fact_6767_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),zero_zero_rat) ).

% fact_not_neg
tff(fact_6768_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),zero_zero_int) ).

% fact_not_neg
tff(fact_6769_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),zero_zero_nat) ).

% fact_not_neg
tff(fact_6770_fact__not__neg,axiom,
    ! [Nb: nat] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),zero_zero_real) ).

% fact_not_neg
tff(fact_6771_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_gt_zero
tff(fact_6772_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_gt_zero
tff(fact_6773_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_gt_zero
tff(fact_6774_fact__gt__zero,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_gt_zero
tff(fact_6775_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L2)),aa_nat_int(semiri1314217659103216013at_int,N2)) ).

% int_sgnE
tff(fact_6776_same__sgn__abs__add,axiom,
    ! [B2: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B2) = aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) )
     => ( abs_abs_Code_integer(plus_p5714425477246183910nteger(A,B2)) = plus_p5714425477246183910nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B2)) ) ) ).

% same_sgn_abs_add
tff(fact_6777_same__sgn__abs__add,axiom,
    ! [B2: real,A: real] :
      ( ( aa_real_real(sgn_sgn_real,B2) = aa_real_real(sgn_sgn_real,A) )
     => ( aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,A)),aa_real_real(abs_abs_real,B2)) ) ) ).

% same_sgn_abs_add
tff(fact_6778_same__sgn__abs__add,axiom,
    ! [B2: rat,A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,B2) = aa_rat_rat(sgn_sgn_rat,A) )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(abs_abs_rat,A)),aa_rat_rat(abs_abs_rat,B2)) ) ) ).

% same_sgn_abs_add
tff(fact_6779_same__sgn__abs__add,axiom,
    ! [B2: int,A: int] :
      ( ( aa_int_int(sgn_sgn_int,B2) = aa_int_int(sgn_sgn_int,A) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2)) ) ) ).

% same_sgn_abs_add
tff(fact_6780_fact__ge__1,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% fact_ge_1
tff(fact_6781_fact__ge__1,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% fact_ge_1
tff(fact_6782_fact__ge__1,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ).

% fact_ge_1
tff(fact_6783_fact__ge__1,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% fact_ge_1
tff(fact_6784_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% fact_mono
tff(fact_6785_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% fact_mono
tff(fact_6786_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% fact_mono
tff(fact_6787_fact__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% fact_mono
tff(fact_6788_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L) )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_6789_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) ) ).

% fact_dvd
tff(fact_6790_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,semiri3624122377584611663nteger(Nb)),semiri3624122377584611663nteger(Mb)) ) ).

% fact_dvd
tff(fact_6791_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) ) ).

% fact_dvd
tff(fact_6792_fact__dvd,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,Mb)) ) ).

% fact_dvd
tff(fact_6793_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(comm_s2602460028002588243omplex(one_one_complex),Nb) ).

% pochhammer_fact
tff(fact_6794_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(comm_s4028243227959126397er_rat(one_one_rat),Nb) ).

% pochhammer_fact
tff(fact_6795_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(comm_s4660882817536571857er_int(one_one_int),Nb) ).

% pochhammer_fact
tff(fact_6796_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,one_one_nat),Nb) ).

% pochhammer_fact
tff(fact_6797_pochhammer__fact,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,one_one_real),Nb) ).

% pochhammer_fact
tff(fact_6798_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_6799_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_6800_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = one_one_real )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% sgn_1_pos
tff(fact_6801_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = one_one_rat )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_6802_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_6803_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_6804_abs__sgn__eq,axiom,
    ! [A: real] :
      aa_real_real(abs_abs_real,aa_real_real(sgn_sgn_real,A)) = $ite(A = zero_zero_real,zero_zero_real,one_one_real) ).

% abs_sgn_eq
tff(fact_6805_abs__sgn__eq,axiom,
    ! [A: rat] :
      aa_rat_rat(abs_abs_rat,aa_rat_rat(sgn_sgn_rat,A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_6806_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_6807_dvd__fact,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Mb),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% dvd_fact
tff(fact_6808_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_nat_rat(semiri773545260158071498ct_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ) ).

% fact_less_mono
tff(fact_6809_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1406184849735516958ct_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ) ).

% fact_less_mono
tff(fact_6810_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ) ).

% fact_less_mono
tff(fact_6811_fact__less__mono,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri2265585572941072030t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ) ).

% fact_less_mono
tff(fact_6812_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)
       => ( aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L)) = aa_int_int(sgn_sgn_int,L) ) ) ) ).

% sgn_mod
tff(fact_6813_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,Mb)) = zero_zero_int ) ) ).

% fact_mod
tff(fact_6814_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( modulo364778990260209775nteger(semiri3624122377584611663nteger(Nb),semiri3624122377584611663nteger(Mb)) = zero_z3403309356797280102nteger ) ) ).

% fact_mod
tff(fact_6815_fact__mod,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,Mb)) = zero_zero_nat ) ) ).

% fact_mod
tff(fact_6816_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,semiri3624122377584611663nteger(K)),semiri3624122377584611663nteger(Nb))),semiri3624122377584611663nteger(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6817_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,Nb))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6818_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,Nb))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6819_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6820_fact__fact__dvd__fact,axiom,
    ! [K: nat,Nb: nat] : aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),Nb))) ).

% fact_fact_dvd_fact
tff(fact_6821_fact__le__power,axiom,
    ! [Nb: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6822_fact__le__power,axiom,
    ! [Nb: nat] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1406184849735516958ct_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6823_fact__le__power,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6824_fact__le__power,axiom,
    ! [Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Nb),Nb))) ).

% fact_le_power
tff(fact_6825_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_6826_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real) )
     => ( ( aa_real_real(arccos,Xb) = aa_real_real(arccos,Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_6827_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_6828_fact__diff__Suc,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(suc,Mb))
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,Mb)),Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_6829_sgn__if,axiom,
    ! [Xb: real] :
      aa_real_real(sgn_sgn_real,Xb) = $ite(
        Xb = zero_zero_real,
        zero_zero_real,
        $ite(aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb),one_one_real,aa_real_real(uminus_uminus_real,one_one_real)) ) ).

% sgn_if
tff(fact_6830_sgn__if,axiom,
    ! [Xb: int] :
      aa_int_int(sgn_sgn_int,Xb) = $ite(
        Xb = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Xb),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_6831_sgn__if,axiom,
    ! [Xb: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,Xb) = $ite(
        Xb = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Xb),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_6832_sgn__if,axiom,
    ! [Xb: rat] :
      aa_rat_rat(sgn_sgn_rat,Xb) = $ite(
        Xb = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb),one_one_rat,aa_rat_rat(uminus_uminus_rat,one_one_rat)) ) ).

% sgn_if
tff(fact_6833_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( aa_real_real(sgn_sgn_real,A) = aa_real_real(uminus_uminus_real,one_one_real) )
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% sgn_1_neg
tff(fact_6834_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_6835_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_6836_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(sgn_sgn_rat,A) = aa_rat_rat(uminus_uminus_rat,one_one_rat) )
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_6837_zsgn__def,axiom,
    ! [I: int] :
      aa_int_int(sgn_sgn_int,I) = $ite(
        I = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_6838_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_6839_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_6840_norm__sgn,axiom,
    ! [Xb: real] :
      aa_real_real(real_V7735802525324610683m_real,aa_real_real(sgn_sgn_real,Xb)) = $ite(Xb = zero_zero_real,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_6841_norm__sgn,axiom,
    ! [Xb: complex] :
      aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(sgn_sgn_complex,Xb)) = $ite(Xb = zero_zero_complex,zero_zero_real,one_one_real) ).

% norm_sgn
tff(fact_6842_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,V)),aa_int_int(abs_abs_int,L))) = aa_int_int(aa_int_fun_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_6843_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(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_int_int(sgn_sgn_int,L))),aa_int_int(aa_int_fun_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_6844_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,semiri3624122377584611663nteger(K)),semiri3624122377584611663nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),semiri3624122377584611663nteger(Nb)) ) ).

% choose_dvd
tff(fact_6845_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_rat_o(aa_rat_fun_rat_o(dvd_dvd_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ) ).

% choose_dvd
tff(fact_6846_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1406184849735516958ct_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ) ).

% choose_dvd
tff(fact_6847_choose__dvd,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(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) ) ).

% choose_dvd
tff(fact_6848_choose__dvd,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => aa_real_o(aa_real_fun_real_o(dvd_dvd_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))),aa_nat_real(semiri2265585572941072030t_real,Nb)) ) ).

% choose_dvd
tff(fact_6849_fact__numeral,axiom,
    ! [K: num] : aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(numeral_numeral_nat,K)) = aa_complex_complex(times_times_complex(numera6690914467698888265omplex(K)),aa_nat_complex(semiri5044797733671781792omplex,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6850_fact__numeral,axiom,
    ! [K: num] : aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(numeral_numeral_nat,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6851_fact__numeral,axiom,
    ! [K: num] : aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(numeral_numeral_nat,K)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,K)),aa_nat_int(semiri1406184849735516958ct_int,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6852_fact__numeral,axiom,
    ! [K: num] : aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,K)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6853_fact__numeral,axiom,
    ! [K: num] : aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(numeral_numeral_nat,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_num_nat(pred_numeral,K))) ).

% fact_numeral
tff(fact_6854_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_6855_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Y)),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_6856_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(arccos,Xb)),aa_real_real(arccos,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_6857_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(arccos,Y)),pi) ) ) ).

% arccos_ubound
tff(fact_6858_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_6859_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Y)),one_one_real)
     => ( aa_real_real(cos_real,aa_real_real(arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_6860_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Theta)),pi)
     => ( aa_real_real(arccos,aa_real_real(cos_real,Theta)) = aa_real_real(abs_abs_real,Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_6861_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_6862_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_6863_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_6864_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_6865_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => ( aa_real_real(sin_real,aa_real_real(arccos,Xb)) != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
tff(fact_6866_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_real(arccos,aa_real_real(cos_real,Xb)) = aa_real_real(uminus_uminus_real,Xb) ) ) ) ).

% arccos_cos2
tff(fact_6867_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ) ).

% arccos_minus
tff(fact_6868_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_complex(semiri5044797733671781792omplex,Mb) = $ite(Mb = zero_zero_nat,one_one_complex,aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6869_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_rat(semiri773545260158071498ct_rat,Mb) = $ite(Mb = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Mb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6870_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_int(semiri1406184849735516958ct_int,Mb) = $ite(Mb = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6871_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = $ite(Mb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6872_fact__num__eq__if,axiom,
    ! [Mb: nat] :
      aa_nat_real(semiri2265585572941072030t_real,Mb) = $ite(Mb = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))) ).

% fact_num_eq_if
tff(fact_6873_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6874_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6875_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6876_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,Nb)),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6877_fact__reduce,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))) ) ) ).

% fact_reduce
tff(fact_6878_pochhammer__same,axiom,
    ! [Nb: nat] : comm_s8582702949713902594nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(Nb)),Nb) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Nb)),semiri3624122377584611663nteger(Nb)) ).

% pochhammer_same
tff(fact_6879_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),Nb) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(semiri5044797733671781792omplex,Nb)) ).

% pochhammer_same
tff(fact_6880_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(semiri773545260158071498ct_rat,Nb)) ).

% pochhammer_same
tff(fact_6881_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,Nb))),Nb) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,one_one_int)),Nb)),aa_nat_int(semiri1406184849735516958ct_int,Nb)) ).

% pochhammer_same
tff(fact_6882_pochhammer__same,axiom,
    ! [Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),Nb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(semiri2265585572941072030t_real,Nb)) ).

% pochhammer_same
tff(fact_6883_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_6884_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_6885_binomial__fact,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)))) ) ) ).

% binomial_fact
tff(fact_6886_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(semiri5044797733671781792omplex,K)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri5044797733671781792omplex,Nb)),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_6887_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri773545260158071498ct_rat,K)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri773545260158071498ct_rat,Nb)),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_6888_fact__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri2265585572941072030t_real,K)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Nb),K))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri2265585572941072030t_real,Nb)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K))) ) ) ).

% fact_binomial
tff(fact_6889_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_6890_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arccos,aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,pi),aa_real_real(arccos,Xb)) ) ) ).

% arccos_minus_abs
tff(fact_6891_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(aa_int_fun_int_int(divide_divide_int,K),L) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_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_6892_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_6893_divide__int__unfold,axiom,
    ! [K: int,Mb: nat,L: int,Nb: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,K)),aa_nat_int(semiri1314217659103216013at_int,Mb))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,Nb))) = $ite(
        ( ( aa_int_int(sgn_sgn_int,L) = zero_zero_int )
        | ( aa_int_int(sgn_sgn_int,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,Mb),Nb)),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Nb),Mb)))))) ) ).

% divide_int_unfold
tff(fact_6894_sin__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(sin_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4),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,X4),aa_nat_nat(suc,zero_zero_nat))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4))) ).

% sin_coeff_def
tff(fact_6895_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_6896_cos__coeff__def,axiom,
    ! [X4: nat] :
      aa_nat_real(cos_coeff,X4) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),X4),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,X4),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_real(semiri2265585572941072030t_real,X4)),zero_zero_real) ).

% cos_coeff_def
tff(fact_6897_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_6898_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_6899_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sgn_sgn_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% zero_le_sgn_iff
tff(fact_6900_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sgn_sgn_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sgn_le_0_iff
tff(fact_6901_norm__ii,axiom,
    aa_complex_real(real_V1022390504157884413omplex,imaginary_unit) = one_one_real ).

% norm_ii
tff(fact_6902_cos__coeff__0,axiom,
    aa_nat_real(cos_coeff,zero_zero_nat) = one_one_real ).

% cos_coeff_0
tff(fact_6903_divide__i,axiom,
    ! [Xb: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),imaginary_unit) = aa_complex_complex(times_times_complex(aa_complex_complex(uminus1482373934393186551omplex,imaginary_unit)),Xb) ).

% divide_i
tff(fact_6904_complex__i__mult__minus,axiom,
    ! [Xb: complex] : aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(times_times_complex(imaginary_unit),Xb)) = aa_complex_complex(uminus1482373934393186551omplex,Xb) ).

% complex_i_mult_minus
tff(fact_6905_i__squared,axiom,
    aa_complex_complex(times_times_complex(imaginary_unit),imaginary_unit) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ).

% i_squared
tff(fact_6906_divide__numeral__i,axiom,
    ! [Zb: complex,Nb: num] : aa_complex_complex(divide1717551699836669952omplex(Zb),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),Zb))),numera6690914467698888265omplex(Nb)) ).

% divide_numeral_i
tff(fact_6907_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_6908_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_6909_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_6910_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_6911_real__sgn__eq,axiom,
    ! [Xb: real] : aa_real_real(sgn_sgn_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),aa_real_real(abs_abs_real,Xb)) ).

% real_sgn_eq
tff(fact_6912_sgn__eq,axiom,
    ! [Zb: complex] : aa_complex_complex(sgn_sgn_complex,Zb) = aa_complex_complex(divide1717551699836669952omplex(Zb),aa_real_complex(real_V4546457046886955230omplex,aa_complex_real(real_V1022390504157884413omplex,Zb))) ).

% sgn_eq
tff(fact_6913_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_6914_i__times__eq__iff,axiom,
    ! [W: complex,Zb: complex] :
      ( ( aa_complex_complex(times_times_complex(imaginary_unit),W) = Zb )
    <=> ( W = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(imaginary_unit),Zb)) ) ) ).

% i_times_eq_iff
tff(fact_6915_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_6916_imaginary__unit_Ocode,axiom,
    imaginary_unit = aa_real_complex(complex2(zero_zero_real),one_one_real) ).

% imaginary_unit.code
tff(fact_6917_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_complex(complex2(Xb),Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
tff(fact_6918_i__mult__Complex,axiom,
    ! [A: real,B2: real] : aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(complex2(A),B2)) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B2)),A) ).

% i_mult_Complex
tff(fact_6919_Complex__mult__i,axiom,
    ! [A: real,B2: real] : aa_complex_complex(times_times_complex(aa_real_complex(complex2(A),B2)),imaginary_unit) = aa_real_complex(complex2(aa_real_real(uminus_uminus_real,B2)),A) ).

% Complex_mult_i
tff(fact_6920_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_6921_sgn__power__injE,axiom,
    ! [A: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,A)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,A)),Nb)) = Xb )
     => ( ( Xb = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,B2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,B2)),Nb)) )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ( A = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_6922_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [F: fun_nat_fun_nat_nat,A: nat,B2: nat,Acc: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,A),B2),Acc) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A),Acc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B2),aa_nat_nat(aa_nat_fun_nat_nat(F,A),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_6923_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [Xb: fun_nat_fun_nat_nat,Xaa: nat,Xba: nat,Xc: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,Xaa),Xba),Xc) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xba),Xaa),Xc,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(Xb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xaa),one_one_nat)),Xba),aa_nat_nat(aa_nat_fun_nat_nat(Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_6924_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_6925_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_6926_Complex__eq,axiom,
    ! [A: real,B2: real] : aa_real_complex(complex2(A),B2) = 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,B2))) ).

% Complex_eq
tff(fact_6927_complex__split__polar,axiom,
    ! [Zb: complex] :
    ? [R3: real,A4: real] : Zb = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R3)),aa_complex_complex(plus_plus_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(cos_real,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_6928_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_6929_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_6930_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero_real )
     => ( aa_real_real(arctan,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Xb)),pi)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),aa_real_real(arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_6931_fact__code,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_6932_fact__code,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_6933_fact__code,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_6934_fact__code,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_6935_fact__code,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb),one_one_nat)) ).

% fact_code
tff(fact_6936_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_6937_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_6938_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_6939_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_6940_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L) = $ite(
        L = zero_zero_int,
        K,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,L)),aa_o_int(zero_n2684676970156552555ol_int,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L))))))) ) ).

% modulo_int_def
tff(fact_6941_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_6942_norm__cis,axiom,
    ! [A: real] : aa_complex_real(real_V1022390504157884413omplex,aa_real_complex(cis,A)) = one_one_real ).

% norm_cis
tff(fact_6943_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_6944_zless__nat__conj,axiom,
    ! [W: int,Zb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Zb))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ) ).

% zless_nat_conj
tff(fact_6945_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_complex(ring_17405671764205052669omplex,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6946_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_real(ring_1_of_int_real,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6947_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_rat(ring_1_of_int_rat,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6948_of__nat__nat__take__bit__eq,axiom,
    ! [Nb: nat,K: int] : aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(bit_se2923211474154528505it_int(Nb),K))) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se2923211474154528505it_int(Nb),K)) ).

% of_nat_nat_take_bit_eq
tff(fact_6949_zero__less__nat__eq,axiom,
    ! [Zb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_int_nat(nat2,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb) ) ).

% zero_less_nat_eq
tff(fact_6950_diff__nat__numeral,axiom,
    ! [V: num,V2: 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,V2)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),aa_num_int(numeral_numeral_int,V2))) ).

% diff_nat_numeral
tff(fact_6951_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) = aa_int_nat(nat2,Y) )
    <=> ( aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_6952_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( aa_int_nat(nat2,Y) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb) )
    <=> ( Y = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_6953_power2__csqrt,axiom,
    ! [Zb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(csqrt,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Zb ).

% power2_csqrt
tff(fact_6954_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb))),A)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,A)) ) ).

% nat_ceiling_le_eq
tff(fact_6955_one__less__nat__eq,axiom,
    ! [Zb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),aa_int_nat(nat2,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),Zb) ) ).

% one_less_nat_eq
tff(fact_6956_nat__numeral__diff__1,axiom,
    ! [V: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_num_nat(numeral_numeral_nat,V)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,V)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_6957_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_6958_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_6959_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb)),aa_int_nat(nat2,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)),A) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_6960_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,Xb: num,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,Xb)),Nb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_6961_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_6962_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_6963_cis__divide,axiom,
    ! [A: real,B2: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(cis,A)),aa_real_complex(cis,B2)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) ).

% cis_divide
tff(fact_6964_nat__zero__as__int,axiom,
    zero_zero_nat = aa_int_nat(nat2,zero_zero_int) ).

% nat_zero_as_int
tff(fact_6965_nat__numeral__as__int,axiom,
    ! [X4: num] : aa_num_nat(numeral_numeral_nat,X4) = aa_int_nat(nat2,aa_num_int(numeral_numeral_int,X4)) ).

% nat_numeral_as_int
tff(fact_6966_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ).

% nat_mono
tff(fact_6967_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_6968_unset__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Mb),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% unset_bit_nat_def
tff(fact_6969_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_6970_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_6971_Arg__correct,axiom,
    ! [Zb: complex] :
      ( ( Zb != zero_zero_complex )
     => ( ( aa_complex_complex(sgn_sgn_complex,Zb) = aa_real_complex(cis,aa_complex_real(arg,Zb)) )
        & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Zb))
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Zb)),pi) ) ) ).

% Arg_correct
tff(fact_6972_cis__Arg__unique,axiom,
    ! [Zb: complex,Xb: real] :
      ( ( aa_complex_complex(sgn_sgn_complex,Zb) = aa_real_complex(cis,Xb) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),Xb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),pi)
         => ( aa_complex_real(arg,Zb) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_6973_cis__mult,axiom,
    ! [A: real,B2: real] : aa_complex_complex(times_times_complex(aa_real_complex(cis,A)),aa_real_complex(cis,B2)) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ).

% cis_mult
tff(fact_6974_nat__mono__iff,axiom,
    ! [Zb: int,W: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Zb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Zb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ) ).

% nat_mono_iff
tff(fact_6975_of__nat__ceiling,axiom,
    ! [R2: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,R2)))) ).

% of_nat_ceiling
tff(fact_6976_of__nat__ceiling,axiom,
    ! [R2: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,R2),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim2889992004027027881ng_rat,R2)))) ).

% of_nat_ceiling
tff(fact_6977_zless__nat__eq__int__zless,axiom,
    ! [Mb: nat,Zb: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_int_nat(nat2,Zb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),Zb) ) ).

% zless_nat_eq_int_zless
tff(fact_6978_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,Xb)),Nb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% nat_le_iff
tff(fact_6979_nat__int__add,axiom,
    ! [A: nat,B2: nat] : aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) ).

% nat_int_add
tff(fact_6980_int__minus,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),aa_nat_int(semiri1314217659103216013at_int,Mb)))) ).

% int_minus
tff(fact_6981_nat__abs__mult__distrib,axiom,
    ! [W: int,Zb: int] : aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Zb))) = 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,Zb))) ).

% nat_abs_mult_distrib
tff(fact_6982_and__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% and_nat_def
tff(fact_6983_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim7802044766580827645g_real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_6984_of__nat__floor,axiom,
    ! [R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),R2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,R2)))),R2) ) ).

% of_nat_floor
tff(fact_6985_of__nat__floor,axiom,
    ! [R2: rat] :
      ( 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,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,R2)))),R2) ) ).

% of_nat_floor
tff(fact_6986_nat__less__eq__zless,axiom,
    ! [W: int,Zb: 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,Zb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),Zb) ) ) ).

% nat_less_eq_zless
tff(fact_6987_nat__le__eq__zle,axiom,
    ! [W: int,Zb: 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),Zb) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,W)),aa_int_nat(nat2,Zb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,W),Zb) ) ) ).

% nat_le_eq_zle
tff(fact_6988_split__nat,axiom,
    ! [P: fun_nat_o,I: int] :
      ( aa_nat_o(P,aa_int_nat(nat2,I))
    <=> ( ! [N4: nat] :
            ( ( I = aa_nat_int(semiri1314217659103216013at_int,N4) )
           => aa_nat_o(P,N4) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_int,I),zero_zero_int)
         => aa_nat_o(P,zero_zero_nat) ) ) ) ).

% split_nat
tff(fact_6989_le__mult__nat__floor,axiom,
    ! [A: real,B2: real] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ).

% le_mult_nat_floor
tff(fact_6990_le__mult__nat__floor,axiom,
    ! [A: rat,B2: rat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,A))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_nat(nat2,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ).

% le_mult_nat_floor
tff(fact_6991_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_6992_nat__add__distrib,axiom,
    ! [Zb: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,Zb),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,Zb)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_add_distrib
tff(fact_6993_nat__mult__distrib,axiom,
    ! [Zb: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Zb),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,Zb)),aa_int_nat(nat2,Z3)) ) ) ).

% nat_mult_distrib
tff(fact_6994_Suc__as__int,axiom,
    ! [X4: nat] : aa_nat_nat(suc,X4) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),one_one_int)) ).

% Suc_as_int
tff(fact_6995_nat__diff__distrib,axiom,
    ! [Z3: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Z3)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z3),Zb)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Zb),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Zb)),aa_int_nat(nat2,Z3)) ) ) ) ).

% nat_diff_distrib
tff(fact_6996_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_6997_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),L)))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_6998_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib
tff(fact_6999_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ).

% nat_div_distrib'
tff(fact_7000_nat__power__eq,axiom,
    ! [Zb: int,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => ( aa_int_nat(nat2,aa_nat_int(power_power_int(Zb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_int_nat(nat2,Zb)),Nb) ) ) ).

% nat_power_eq
tff(fact_7001_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = zero_zero_nat ) ) ).

% nat_floor_neg
tff(fact_7002_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_int_nat(nat2,Xb)),aa_int_nat(nat2,Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_7003_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_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_7004_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_7005_le__nat__floor,axiom,
    ! [Xb: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Xb)),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,A))) ) ).

% le_nat_floor
tff(fact_7006_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa_int_int(aa_int_fun_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_7007_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_7008_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_7009_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_7010_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_7011_Suc__nat__eq__nat__zadd1,axiom,
    ! [Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Zb)
     => ( aa_nat_nat(suc,aa_int_nat(nat2,Zb)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Zb)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_7012_nat__less__iff,axiom,
    ! [W: int,Mb: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),W)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(nat2,W)),Mb)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,W),aa_nat_int(semiri1314217659103216013at_int,Mb)) ) ) ).

% nat_less_iff
tff(fact_7013_nat__mult__distrib__neg,axiom,
    ! [Zb: int,Z3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),zero_zero_int)
     => ( aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,Zb),Z3)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Zb))),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Z3))) ) ) ).

% nat_mult_distrib_neg
tff(fact_7014_nat__abs__int__diff,axiom,
    ! [A: nat,B2: nat] :
      aa_int_nat(nat2,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,A)),aa_nat_int(semiri1314217659103216013at_int,B2)))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B2),A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) ).

% nat_abs_int_diff
tff(fact_7015_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Nb)))
       => ( aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_7016_cis__conv__exp,axiom,
    ! [B2: real] : aa_real_complex(cis,B2) = aa_complex_complex(exp_complex,aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,B2))) ).

% cis_conv_exp
tff(fact_7017_of__int__of__nat,axiom,
    ! [K: int] :
      ring_18347121197199848620nteger(K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,semiri4939895301339042750nteger(aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),semiri4939895301339042750nteger(aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7018_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_complex(semiri8010041392384452111omplex,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7019_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_real(semiri5074537144036343181t_real,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7020_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_rat(ring_1_of_int_rat,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_rat(semiri681578069525770553at_rat,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7021_of__int__of__nat,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,aa_int_int(uminus_uminus_int,K)))),aa_nat_int(semiri1314217659103216013at_int,aa_int_nat(nat2,K))) ).

% of_int_of_nat
tff(fact_7022_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,Xb)) = aa_complex_complex(csqrt,aa_real_complex(real_V4546457046886955230omplex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_7023_Arg__bounded,axiom,
    ! [Zb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,pi)),aa_complex_real(arg,Zb))
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(arg,Zb)),pi) ) ).

% Arg_bounded
tff(fact_7024_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_7025_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,I)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,I)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,I))))) ) ) ).

% powr_int
tff(fact_7026_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L) = $ite(
        L = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_int(sgn_sgn_int,K) = aa_int_int(sgn_sgn_int,L),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),aa_int_int(uminus_uminus_int,aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),aa_int_nat(nat2,aa_int_int(abs_abs_int,L)))),zero_n2687167440665602831ol_nat(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),K)))))) ) ).

% divide_int_def
tff(fact_7027_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_complex ) ) ).

% cis_multiple_2pi
tff(fact_7028_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,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,Xb),aa_int_nat(nat2,Nb)),aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_int_nat(nat2,aa_int_int(uminus_uminus_int,Nb))))) ) ) ).

% powr_real_of_int
tff(fact_7029_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_xor_eq
tff(fact_7030_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% xor_Suc_0_eq
tff(fact_7031_gbinomial__absorption_H,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7032_gbinomial__absorption_H,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7033_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))) ) ) ).

% gbinomial_absorption'
tff(fact_7034_inverse__inverse__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ).

% inverse_inverse_eq
tff(fact_7035_inverse__inverse__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ).

% inverse_inverse_eq
tff(fact_7036_inverse__inverse__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ).

% inverse_inverse_eq
tff(fact_7037_inverse__eq__iff__eq,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B2) )
    <=> ( A = B2 ) ) ).

% inverse_eq_iff_eq
tff(fact_7038_inverse__eq__iff__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B2) )
    <=> ( A = B2 ) ) ).

% inverse_eq_iff_eq
tff(fact_7039_inverse__eq__iff__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B2) )
    <=> ( A = B2 ) ) ).

% inverse_eq_iff_eq
tff(fact_7040_bit_Oxor__left__self,axiom,
    ! [Xb: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),aa_int_int(bit_se6526347334894502574or_int(Xb),Y)) = Y ).

% bit.xor_left_self
tff(fact_7041_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
    <=> ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7042_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
    <=> ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7043_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_7044_inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% inverse_zero
tff(fact_7045_inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% inverse_zero
tff(fact_7046_inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% inverse_zero
tff(fact_7047_inverse__mult__distrib,axiom,
    ! [A: real,B2: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) ).

% inverse_mult_distrib
tff(fact_7048_inverse__mult__distrib,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) ).

% inverse_mult_distrib
tff(fact_7049_inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) ).

% inverse_mult_distrib
tff(fact_7050_inverse__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(inverse_inverse_real,Xb) = one_one_real )
    <=> ( Xb = one_one_real ) ) ).

% inverse_eq_1_iff
tff(fact_7051_inverse__eq__1__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,Xb) = one_one_complex )
    <=> ( Xb = one_one_complex ) ) ).

% inverse_eq_1_iff
tff(fact_7052_inverse__eq__1__iff,axiom,
    ! [Xb: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,Xb) = one_one_rat )
    <=> ( Xb = one_one_rat ) ) ).

% inverse_eq_1_iff
tff(fact_7053_inverse__1,axiom,
    aa_real_real(inverse_inverse_real,one_one_real) = one_one_real ).

% inverse_1
tff(fact_7054_inverse__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,one_one_complex) = one_one_complex ).

% inverse_1
tff(fact_7055_inverse__1,axiom,
    aa_rat_rat(inverse_inverse_rat,one_one_rat) = one_one_rat ).

% inverse_1
tff(fact_7056_inverse__divide,axiom,
    ! [A: real,B2: real] : aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,B2),A) ).

% inverse_divide
tff(fact_7057_inverse__divide,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = aa_complex_complex(divide1717551699836669952omplex(B2),A) ).

% inverse_divide
tff(fact_7058_inverse__divide,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(divide_divide_rat(A),B2)) = aa_rat_rat(divide_divide_rat(B2),A) ).

% inverse_divide
tff(fact_7059_inverse__minus__eq,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ).

% inverse_minus_eq
tff(fact_7060_inverse__minus__eq,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ).

% inverse_minus_eq
tff(fact_7061_inverse__minus__eq,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ).

% inverse_minus_eq
tff(fact_7062_xor_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),zero_zero_nat) = A ).

% xor.right_neutral
tff(fact_7063_xor_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),zero_zero_int) = A ).

% xor.right_neutral
tff(fact_7064_xor_Oleft__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,zero_zero_nat),A) = A ).

% xor.left_neutral
tff(fact_7065_xor_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(zero_zero_int),A) = A ).

% xor.left_neutral
tff(fact_7066_xor__self__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),A) = zero_zero_nat ).

% xor_self_eq
tff(fact_7067_xor__self__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),A) = zero_zero_int ).

% xor_self_eq
tff(fact_7068_bit_Oxor__self,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),Xb) = zero_zero_int ).

% bit.xor_self
tff(fact_7069_abs__inverse,axiom,
    ! [A: real] : aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ).

% abs_inverse
tff(fact_7070_abs__inverse,axiom,
    ! [A: complex] : aa_complex_complex(abs_abs_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(abs_abs_complex,A)) ).

% abs_inverse
tff(fact_7071_abs__inverse,axiom,
    ! [A: rat] : aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ).

% abs_inverse
tff(fact_7072_inverse__sgn,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) = aa_real_real(sgn_sgn_real,A) ).

% inverse_sgn
tff(fact_7073_inverse__sgn,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) = aa_rat_rat(sgn_sgn_rat,A) ).

% inverse_sgn
tff(fact_7074_sgn__inverse,axiom,
    ! [A: real] : aa_real_real(sgn_sgn_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(sgn_sgn_real,A)) ).

% sgn_inverse
tff(fact_7075_sgn__inverse,axiom,
    ! [A: complex] : aa_complex_complex(sgn_sgn_complex,aa_complex_complex(invers8013647133539491842omplex,A)) = aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(sgn_sgn_complex,A)) ).

% sgn_inverse
tff(fact_7076_sgn__inverse,axiom,
    ! [A: rat] : aa_rat_rat(sgn_sgn_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(sgn_sgn_rat,A)) ).

% sgn_inverse
tff(fact_7077_take__bit__xor,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_se6526347334894502574or_int(A),B2)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se2923211474154528505it_int(Nb),A)),aa_int_int(bit_se2923211474154528505it_int(Nb),B2)) ).

% take_bit_xor
tff(fact_7078_take__bit__xor,axiom,
    ! [Nb: nat,A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_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),B2)) ).

% take_bit_xor
tff(fact_7079_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),zero_zero_real) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7080_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),zero_zero_rat) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_7081_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7082_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_7083_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ).

% inverse_positive_iff_positive
tff(fact_7084_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ).

% inverse_positive_iff_positive
tff(fact_7085_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ).

% inverse_negative_iff_negative
tff(fact_7086_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ).

% inverse_negative_iff_negative
tff(fact_7087_inverse__less__iff__less__neg,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7088_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_7089_inverse__less__iff__less,axiom,
    ! [A: real,B2: 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),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7090_inverse__less__iff__less,axiom,
    ! [A: rat,B2: 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),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ) ).

% inverse_less_iff_less
tff(fact_7091_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,zero_zero_real),aa_nat_nat(suc,K)) = zero_zero_real ).

% gbinomial_0(2)
tff(fact_7092_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_rat(gbinomial_rat(zero_zero_rat),aa_nat_nat(suc,K)) = zero_zero_rat ).

% gbinomial_0(2)
tff(fact_7093_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,zero_zero_nat),aa_nat_nat(suc,K)) = zero_zero_nat ).

% gbinomial_0(2)
tff(fact_7094_gbinomial__0_I2_J,axiom,
    ! [K: nat] : aa_nat_int(gbinomial_int(zero_zero_int),aa_nat_nat(suc,K)) = zero_zero_int ).

% gbinomial_0(2)
tff(fact_7095_gbinomial__0_I1_J,axiom,
    ! [A: complex] : aa_nat_complex(gbinomial_complex(A),zero_zero_nat) = one_one_complex ).

% gbinomial_0(1)
tff(fact_7096_gbinomial__0_I1_J,axiom,
    ! [A: real] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),zero_zero_nat) = one_one_real ).

% gbinomial_0(1)
tff(fact_7097_gbinomial__0_I1_J,axiom,
    ! [A: rat] : aa_nat_rat(gbinomial_rat(A),zero_zero_nat) = one_one_rat ).

% gbinomial_0(1)
tff(fact_7098_gbinomial__0_I1_J,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),zero_zero_nat) = one_one_nat ).

% gbinomial_0(1)
tff(fact_7099_gbinomial__0_I1_J,axiom,
    ! [A: int] : aa_nat_int(gbinomial_int(A),zero_zero_nat) = one_one_int ).

% gbinomial_0(1)
tff(fact_7100_inverse__le__iff__le,axiom,
    ! [A: real,B2: 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),B2)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7101_inverse__le__iff__le,axiom,
    ! [A: rat,B2: 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),B2)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ) ).

% inverse_le_iff_le
tff(fact_7102_inverse__le__iff__le__neg,axiom,
    ! [A: real,B2: 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,B2),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7103_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B2: 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,B2),zero_zero_rat)
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_7104_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,A)) = one_one_real ) ) ).

% right_inverse
tff(fact_7105_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,A)) = one_one_complex ) ) ).

% right_inverse
tff(fact_7106_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,A)) = one_one_rat ) ) ).

% right_inverse
tff(fact_7107_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% left_inverse
tff(fact_7108_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% left_inverse
tff(fact_7109_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% left_inverse
tff(fact_7110_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,W)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_num_real(numeral_numeral_real,W)) ).

% inverse_eq_divide_numeral
tff(fact_7111_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(W)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(W)) ).

% inverse_eq_divide_numeral
tff(fact_7112_inverse__eq__divide__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,W)) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,W)) ).

% inverse_eq_divide_numeral
tff(fact_7113_floor__add2,axiom,
    ! [Xb: real,Y: real] :
      ( ( member_real(Xb,ring_1_Ints_real)
        | member_real(Y,ring_1_Ints_real) )
     => ( aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_real_int(archim6058952711729229775r_real,Xb)),aa_real_int(archim6058952711729229775r_real,Y)) ) ) ).

% floor_add2
tff(fact_7114_floor__add2,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( member_rat(Xb,ring_1_Ints_rat)
        | member_rat(Y,ring_1_Ints_rat) )
     => ( aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_rat_int(archim3151403230148437115or_rat,Xb)),aa_rat_int(archim3151403230148437115or_rat,Y)) ) ) ).

% floor_add2
tff(fact_7115_frac__gt__0__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(archim2898591450579166408c_real,Xb))
    <=> ~ member_real(Xb,ring_1_Ints_real) ) ).

% frac_gt_0_iff
tff(fact_7116_frac__gt__0__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(archimedean_frac_rat,Xb))
    <=> ~ member_rat(Xb,ring_1_Ints_rat) ) ).

% frac_gt_0_iff
tff(fact_7117_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(uminus_uminus_real,aa_num_real(numeral_numeral_real,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7118_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(uminus1482373934393186551omplex,numera6690914467698888265omplex(W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7119_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] : aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_rat_rat(uminus_uminus_rat,aa_num_rat(numeral_numeral_rat,W))) ).

% inverse_eq_divide_neg_numeral
tff(fact_7120_xor__numerals_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(3)
tff(fact_7121_xor__numerals_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).

% xor_numerals(3)
tff(fact_7122_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_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)) ).

% xor_numerals(1)
tff(fact_7123_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_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)) ).

% xor_numerals(1)
tff(fact_7124_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Y))) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_7125_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y)) ).

% xor_numerals(2)
tff(fact_7126_xor__numerals_I5_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_numerals(5)
tff(fact_7127_xor__numerals_I5_J,axiom,
    ! [Xb: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb)) ).

% xor_numerals(5)
tff(fact_7128_xor__numerals_I8_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),one_one_nat) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_numerals(8)
tff(fact_7129_xor__numerals_I8_J,axiom,
    ! [Xb: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),one_one_int) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb)) ).

% xor_numerals(8)
tff(fact_7130_xor__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y))) ).

% xor_numerals(7)
tff(fact_7131_xor__numerals_I7_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y))) ).

% xor_numerals(7)
tff(fact_7132_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_7133_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_7134_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_7135_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb)) ).

% xor_nat_numerals(4)
tff(fact_7136_xor__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),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_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(4)
tff(fact_7137_xor__numerals_I4_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit1,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(4)
tff(fact_7138_xor__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),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_se6528837805403552850or_nat,aa_num_nat(numeral_numeral_nat,Xb)),aa_num_nat(numeral_numeral_nat,Y)))) ).

% xor_numerals(6)
tff(fact_7139_xor__numerals_I6_J,axiom,
    ! [Xb: num,Y: num] : aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Xb))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,Y))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_num_int(numeral_numeral_int,Xb)),aa_num_int(numeral_numeral_int,Y)))) ).

% xor_numerals(6)
tff(fact_7140_power__inverse,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,A)),Nb) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb)) ).

% power_inverse
tff(fact_7141_power__inverse,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,A)),Nb) = aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(power_power_complex(A),Nb)) ).

% power_inverse
tff(fact_7142_power__inverse,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,A)),Nb) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(power_power_rat(A),Nb)) ).

% power_inverse
tff(fact_7143_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Y),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y) )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Y)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7144_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,Xb: complex] :
      ( ( aa_complex_complex(times_times_complex(Y),Xb) = aa_complex_complex(times_times_complex(Xb),Y) )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Y)),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7145_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: rat,Xb: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Y)),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,Y)) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_7146_of__nat__xor__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(semiri1316708129612266289at_nat,Mb)),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_xor_eq
tff(fact_7147_of__nat__xor__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb)) = aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_xor_eq
tff(fact_7148_of__int__xor__eq,axiom,
    ! [K: int,L: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_se6526347334894502574or_int(K),L)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(ring_1_of_int_int,K)),aa_int_int(ring_1_of_int_int,L)) ).

% of_int_xor_eq
tff(fact_7149_xor_Oleft__commute,axiom,
    ! [B2: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B2),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B2),C)) ).

% xor.left_commute
tff(fact_7150_xor_Oleft__commute,axiom,
    ! [B2: int,A: int,C: int] : aa_int_int(bit_se6526347334894502574or_int(B2),aa_int_int(bit_se6526347334894502574or_int(A),C)) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se6526347334894502574or_int(B2),C)) ).

% xor.left_commute
tff(fact_7151_xor_Ocommute,axiom,
    ! [A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B2),A) ).

% xor.commute
tff(fact_7152_xor_Ocommute,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_se6526347334894502574or_int(A),B2) = aa_int_int(bit_se6526347334894502574or_int(B2),A) ).

% xor.commute
tff(fact_7153_xor_Oassoc,axiom,
    ! [A: nat,B2: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B2)),C) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,B2),C)) ).

% xor.assoc
tff(fact_7154_xor_Oassoc,axiom,
    ! [A: int,B2: int,C: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se6526347334894502574or_int(A),B2)),C) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se6526347334894502574or_int(B2),C)) ).

% xor.assoc
tff(fact_7155_inverse__eq__imp__eq,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B2) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
tff(fact_7156_inverse__eq__imp__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B2) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
tff(fact_7157_inverse__eq__imp__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B2) )
     => ( A = B2 ) ) ).

% inverse_eq_imp_eq
tff(fact_7158_field__class_Ofield__inverse__zero,axiom,
    aa_real_real(inverse_inverse_real,zero_zero_real) = zero_zero_real ).

% field_class.field_inverse_zero
tff(fact_7159_field__class_Ofield__inverse__zero,axiom,
    aa_complex_complex(invers8013647133539491842omplex,zero_zero_complex) = zero_zero_complex ).

% field_class.field_inverse_zero
tff(fact_7160_field__class_Ofield__inverse__zero,axiom,
    aa_rat_rat(inverse_inverse_rat,zero_zero_rat) = zero_zero_rat ).

% field_class.field_inverse_zero
tff(fact_7161_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
tff(fact_7162_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
tff(fact_7163_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
tff(fact_7164_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(inverse_inverse_real,A) = aa_real_real(inverse_inverse_real,B2) )
     => ( ( A != zero_zero_real )
       => ( ( B2 != zero_zero_real )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7165_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(invers8013647133539491842omplex,B2) )
     => ( ( A != zero_zero_complex )
       => ( ( B2 != zero_zero_complex )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7166_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(inverse_inverse_rat,B2) )
     => ( ( A != zero_zero_rat )
       => ( ( B2 != zero_zero_rat )
         => ( A = B2 ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_7167_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(inverse_inverse_real,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7168_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(invers8013647133539491842omplex,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7169_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(inverse_inverse_rat,A)) = A ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_7170_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7171_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7172_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_7173_Ints__numeral,axiom,
    ! [Nb: num] : member_complex(numera6690914467698888265omplex(Nb),ring_1_Ints_complex) ).

% Ints_numeral
tff(fact_7174_Ints__numeral,axiom,
    ! [Nb: num] : member_real(aa_num_real(numeral_numeral_real,Nb),ring_1_Ints_real) ).

% Ints_numeral
tff(fact_7175_Ints__numeral,axiom,
    ! [Nb: num] : member_rat(aa_num_rat(numeral_numeral_rat,Nb),ring_1_Ints_rat) ).

% Ints_numeral
tff(fact_7176_Ints__numeral,axiom,
    ! [Nb: num] : member_int(aa_num_int(numeral_numeral_int,Nb),ring_1_Ints_int) ).

% Ints_numeral
tff(fact_7177_Ints__mult,axiom,
    ! [A: complex,B2: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B2,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(times_times_complex(A),B2),ring_1_Ints_complex) ) ) ).

% Ints_mult
tff(fact_7178_Ints__mult,axiom,
    ! [A: real,B2: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B2,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2),ring_1_Ints_real) ) ) ).

% Ints_mult
tff(fact_7179_Ints__mult,axiom,
    ! [A: rat,B2: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B2,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2),ring_1_Ints_rat) ) ) ).

% Ints_mult
tff(fact_7180_Ints__mult,axiom,
    ! [A: int,B2: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B2,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B2),ring_1_Ints_int) ) ) ).

% Ints_mult
tff(fact_7181_Ints__1,axiom,
    member_complex(one_one_complex,ring_1_Ints_complex) ).

% Ints_1
tff(fact_7182_Ints__1,axiom,
    member_rat(one_one_rat,ring_1_Ints_rat) ).

% Ints_1
tff(fact_7183_Ints__1,axiom,
    member_int(one_one_int,ring_1_Ints_int) ).

% Ints_1
tff(fact_7184_Ints__1,axiom,
    member_real(one_one_real,ring_1_Ints_real) ).

% Ints_1
tff(fact_7185_Ints__add,axiom,
    ! [A: complex,B2: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B2,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(plus_plus_complex(A),B2),ring_1_Ints_complex) ) ) ).

% Ints_add
tff(fact_7186_Ints__add,axiom,
    ! [A: real,B2: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B2,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2),ring_1_Ints_real) ) ) ).

% Ints_add
tff(fact_7187_Ints__add,axiom,
    ! [A: rat,B2: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B2,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2),ring_1_Ints_rat) ) ) ).

% Ints_add
tff(fact_7188_Ints__add,axiom,
    ! [A: int,B2: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B2,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2),ring_1_Ints_int) ) ) ).

% Ints_add
tff(fact_7189_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa_real_real(sqrt,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_7190_Ints__diff,axiom,
    ! [A: complex,B2: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( member_complex(B2,ring_1_Ints_complex)
       => member_complex(aa_complex_complex(minus_minus_complex(A),B2),ring_1_Ints_complex) ) ) ).

% Ints_diff
tff(fact_7191_Ints__diff,axiom,
    ! [A: real,B2: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( member_real(B2,ring_1_Ints_real)
       => member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2),ring_1_Ints_real) ) ) ).

% Ints_diff
tff(fact_7192_Ints__diff,axiom,
    ! [A: rat,B2: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( member_rat(B2,ring_1_Ints_rat)
       => member_rat(aa_rat_rat(minus_minus_rat(A),B2),ring_1_Ints_rat) ) ) ).

% Ints_diff
tff(fact_7193_Ints__diff,axiom,
    ! [A: int,B2: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( member_int(B2,ring_1_Ints_int)
       => member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2),ring_1_Ints_int) ) ) ).

% Ints_diff
tff(fact_7194_Ints__power,axiom,
    ! [A: real,Nb: nat] :
      ( member_real(A,ring_1_Ints_real)
     => member_real(aa_nat_real(aa_real_fun_nat_real(power_power_real,A),Nb),ring_1_Ints_real) ) ).

% Ints_power
tff(fact_7195_Ints__power,axiom,
    ! [A: int,Nb: nat] :
      ( member_int(A,ring_1_Ints_int)
     => member_int(aa_nat_int(power_power_int(A),Nb),ring_1_Ints_int) ) ).

% Ints_power
tff(fact_7196_Ints__power,axiom,
    ! [A: complex,Nb: nat] :
      ( member_complex(A,ring_1_Ints_complex)
     => member_complex(aa_nat_complex(power_power_complex(A),Nb),ring_1_Ints_complex) ) ).

% Ints_power
tff(fact_7197_bit_Oconj__xor__distrib2,axiom,
    ! [Y: int,Zb: int,Xb: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se6526347334894502574or_int(Y),Zb)),Xb) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se725231765392027082nd_int(Y),Xb)),aa_int_int(bit_se725231765392027082nd_int(Zb),Xb)) ).

% bit.conj_xor_distrib2
tff(fact_7198_bit_Oconj__xor__distrib,axiom,
    ! [Xb: int,Y: int,Zb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),aa_int_int(bit_se6526347334894502574or_int(Y),Zb)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),aa_int_int(bit_se725231765392027082nd_int(Xb),Zb)) ).

% bit.conj_xor_distrib
tff(fact_7199_bit__xor__iff,axiom,
    ! [A: nat,B2: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B2)),Nb)
    <=> ~ ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
        <=> aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,B2),Nb) ) ) ).

% bit_xor_iff
tff(fact_7200_bit__xor__iff,axiom,
    ! [A: int,B2: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se6526347334894502574or_int(A),B2)),Nb)
    <=> ~ ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
        <=> aa_nat_o(bit_se1146084159140164899it_int(B2),Nb) ) ) ).

% bit_xor_iff
tff(fact_7201_norm__inverse__le__norm,axiom,
    ! [R2: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_real_real(real_V7735802525324610683m_real,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(inverse_inverse_real,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_7202_norm__inverse__le__norm,axiom,
    ! [R2: real,Xb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,R2),aa_complex_real(real_V1022390504157884413omplex,Xb))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R2)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(invers8013647133539491842omplex,Xb))),aa_real_real(inverse_inverse_real,R2)) ) ) ).

% norm_inverse_le_norm
tff(fact_7203_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7204_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ).

% positive_imp_inverse_positive
tff(fact_7205_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real) ) ).

% negative_imp_inverse_negative
tff(fact_7206_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat) ) ).

% negative_imp_inverse_negative
tff(fact_7207_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,A))
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7208_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(inverse_inverse_rat,A))
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A) ) ) ).

% inverse_positive_imp_positive
tff(fact_7209_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),zero_zero_real)
     => ( ( A != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% inverse_negative_imp_negative
tff(fact_7210_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),zero_zero_rat)
     => ( ( A != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% inverse_negative_imp_negative
tff(fact_7211_less__imp__inverse__less__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7212_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less_neg
tff(fact_7213_inverse__less__imp__less__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7214_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ) ).

% inverse_less_imp_less_neg
tff(fact_7215_less__imp__inverse__less,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7216_less__imp__inverse__less,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% less_imp_inverse_less
tff(fact_7217_inverse__less__imp__less,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( 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,B2),A) ) ) ).

% inverse_less_imp_less
tff(fact_7218_inverse__less__imp__less,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( 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,B2),A) ) ) ).

% inverse_less_imp_less
tff(fact_7219_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7220_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(times_times_complex(A),B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B2)),aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7221_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_7222_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,A)) = aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7223_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(uminus1482373934393186551omplex,A)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(invers8013647133539491842omplex,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7224_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,aa_rat_rat(uminus_uminus_rat,A)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_7225_inverse__numeral__1,axiom,
    aa_real_real(inverse_inverse_real,aa_num_real(numeral_numeral_real,one)) = aa_num_real(numeral_numeral_real,one) ).

% inverse_numeral_1
tff(fact_7226_inverse__numeral__1,axiom,
    aa_complex_complex(invers8013647133539491842omplex,numera6690914467698888265omplex(one)) = numera6690914467698888265omplex(one) ).

% inverse_numeral_1
tff(fact_7227_inverse__numeral__1,axiom,
    aa_rat_rat(inverse_inverse_rat,aa_num_rat(numeral_numeral_rat,one)) = aa_num_rat(numeral_numeral_rat,one) ).

% inverse_numeral_1
tff(fact_7228_inverse__unique,axiom,
    ! [A: real,B2: real] :
      ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2) = one_one_real )
     => ( aa_real_real(inverse_inverse_real,A) = B2 ) ) ).

% inverse_unique
tff(fact_7229_inverse__unique,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_complex(times_times_complex(A),B2) = one_one_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = B2 ) ) ).

% inverse_unique
tff(fact_7230_inverse__unique,axiom,
    ! [A: rat,B2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2) = one_one_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = B2 ) ) ).

% inverse_unique
tff(fact_7231_divide__inverse__commute,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,B2)),A) ).

% divide_inverse_commute
tff(fact_7232_divide__inverse__commute,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,B2)),A) ).

% divide_inverse_commute
tff(fact_7233_divide__inverse__commute,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(divide_divide_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,B2)),A) ).

% divide_inverse_commute
tff(fact_7234_divide__inverse,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B2)) ).

% divide_inverse
tff(fact_7235_divide__inverse,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B2)) ).

% divide_inverse
tff(fact_7236_divide__inverse,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(divide_divide_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B2)) ).

% divide_inverse
tff(fact_7237_field__class_Ofield__divide__inverse,axiom,
    ! [A: real,B2: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),B2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_real_real(inverse_inverse_real,B2)) ).

% field_class.field_divide_inverse
tff(fact_7238_field__class_Ofield__divide__inverse,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(times_times_complex(A),aa_complex_complex(invers8013647133539491842omplex,B2)) ).

% field_class.field_divide_inverse
tff(fact_7239_field__class_Ofield__divide__inverse,axiom,
    ! [A: rat,B2: rat] : aa_rat_rat(divide_divide_rat(A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(inverse_inverse_rat,B2)) ).

% field_class.field_divide_inverse
tff(fact_7240_inverse__eq__divide,axiom,
    ! [A: real] : aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ).

% inverse_eq_divide
tff(fact_7241_inverse__eq__divide,axiom,
    ! [A: complex] : aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ).

% inverse_eq_divide
tff(fact_7242_inverse__eq__divide,axiom,
    ! [A: rat] : aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ).

% inverse_eq_divide
tff(fact_7243_power__mult__power__inverse__commute,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_7244_power__mult__power__inverse__commute,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Nb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_7245_power__mult__power__inverse__commute,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Nb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_power_inverse_commute
tff(fact_7246_power__mult__inverse__distrib,axiom,
    ! [Xb: real,Mb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,Xb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_7247_power__mult__inverse__distrib,axiom,
    ! [Xb: complex,Mb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Mb)),aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),aa_nat_complex(power_power_complex(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_7248_power__mult__inverse__distrib,axiom,
    ! [Xb: rat,Mb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Mb)),aa_rat_rat(inverse_inverse_rat,Xb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,Xb)),aa_nat_rat(power_power_rat(Xb),Mb)) ).

% power_mult_inverse_distrib
tff(fact_7249_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_7250_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_nat_complex(semiri8010041392384452111omplex,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_7251_mult__inverse__of__nat__commute,axiom,
    ! [Xaa: nat,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,Xaa))) ).

% mult_inverse_of_nat_commute
tff(fact_7252_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(abs_abs_real,aa_real_real(inverse_inverse_real,A)) = aa_real_real(inverse_inverse_real,aa_real_real(abs_abs_real,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7253_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(abs_abs_rat,aa_rat_rat(inverse_inverse_rat,A)) = aa_rat_rat(inverse_inverse_rat,aa_rat_rat(abs_abs_rat,A)) ) ) ).

% nonzero_abs_inverse
tff(fact_7254_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,aa_int_real(ring_1_of_int_real,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_7255_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: complex] : aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))),Xb) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,aa_int_complex(ring_17405671764205052669omplex,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_7256_mult__inverse__of__int__commute,axiom,
    ! [Xaa: int,Xb: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))),Xb) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),aa_rat_rat(inverse_inverse_rat,aa_int_rat(ring_1_of_int_rat,Xaa))) ).

% mult_inverse_of_int_commute
tff(fact_7257_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),aa_real_real(inverse_inverse_real,Y)) ).

% divide_real_def
tff(fact_7258_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( ( aa_complex_complex(plus_plus_complex(A),A) = zero_zero_complex )
      <=> ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7259_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),A) = zero_zero_real )
      <=> ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7260_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) = zero_zero_rat )
      <=> ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7261_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_7262_inverse__le__imp__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( 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,B2),A) ) ) ).

% inverse_le_imp_le
tff(fact_7263_inverse__le__imp__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( 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,B2),A) ) ) ).

% inverse_le_imp_le
tff(fact_7264_le__imp__inverse__le,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7265_le__imp__inverse__le,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le
tff(fact_7266_inverse__le__imp__le__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7267_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% inverse_le_imp_le_neg
tff(fact_7268_le__imp__inverse__le__neg,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),zero_zero_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,B2)),aa_real_real(inverse_inverse_real,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7269_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),zero_zero_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,B2)),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% le_imp_inverse_le_neg
tff(fact_7270_inverse__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_7271_inverse__le__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),Xb) ) ) ).

% inverse_le_1_iff
tff(fact_7272_one__less__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% one_less_inverse_iff
tff(fact_7273_one__less__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),one_one_rat) ) ) ).

% one_less_inverse_iff
tff(fact_7274_one__less__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_less_inverse
tff(fact_7275_one__less__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_less_inverse
tff(fact_7276_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),A) = one_one_real ) ) ).

% field_class.field_inverse
tff(fact_7277_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),A) = one_one_complex ) ) ).

% field_class.field_inverse
tff(fact_7278_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),A) = one_one_rat ) ) ).

% field_class.field_inverse
tff(fact_7279_division__ring__inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7280_division__ring__inverse__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(plus_plus_complex(A),B2))),aa_complex_complex(invers8013647133539491842omplex,B2)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7281_division__ring__inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% division_ring_inverse_add
tff(fact_7282_inverse__add,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)),aa_real_real(inverse_inverse_real,A))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% inverse_add
tff(fact_7283_inverse__add,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),B2)),aa_complex_complex(invers8013647133539491842omplex,A))),aa_complex_complex(invers8013647133539491842omplex,B2)) ) ) ) ).

% inverse_add
tff(fact_7284_inverse__add,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2)),aa_rat_rat(inverse_inverse_rat,A))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% inverse_add
tff(fact_7285_division__ring__inverse__diff,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A))),aa_real_real(inverse_inverse_real,B2)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7286_division__ring__inverse__diff,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(B2),A))),aa_complex_complex(invers8013647133539491842omplex,B2)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7287_division__ring__inverse__diff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(B2),A))),aa_rat_rat(inverse_inverse_rat,B2)) ) ) ) ).

% division_ring_inverse_diff
tff(fact_7288_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( aa_real_real(inverse_inverse_real,A) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7289_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( aa_complex_complex(invers8013647133539491842omplex,A) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7290_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(inverse_inverse_rat,A) = aa_rat_rat(divide_divide_rat(one_one_rat),A) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_7291_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7292_gbinomial__Suc__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7293_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) ).

% gbinomial_Suc_Suc
tff(fact_7294_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),K) = aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7295_gbinomial__of__nat__symmetric,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(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),K) = aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7296_gbinomial__of__nat__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Nb)
     => ( aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),K) = aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Nb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),K)) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_7297_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_7298_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( member_complex(A,ring_1_Ints_complex)
     => ( aa_complex_complex(plus_plus_complex(aa_complex_complex(plus_plus_complex(one_one_complex),A)),A) != zero_zero_complex ) ) ).

% Ints_odd_nonzero
tff(fact_7299_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A) != zero_zero_real ) ) ).

% Ints_odd_nonzero
tff(fact_7300_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A) != zero_zero_rat ) ) ).

% Ints_odd_nonzero
tff(fact_7301_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A) != zero_zero_int ) ) ).

% Ints_odd_nonzero
tff(fact_7302_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => member_complex(aa_complex_complex(divide1717551699836669952omplex(aa_int_complex(ring_17405671764205052669omplex,A)),aa_int_complex(ring_17405671764205052669omplex,B2)),ring_1_Ints_complex) ) ).

% of_int_divide_in_Ints
tff(fact_7303_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => member_real(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,A)),aa_int_real(ring_1_of_int_real,B2)),ring_1_Ints_real) ) ).

% of_int_divide_in_Ints
tff(fact_7304_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => member_rat(aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,A)),aa_int_rat(ring_1_of_int_rat,B2)),ring_1_Ints_rat) ) ).

% of_int_divide_in_Ints
tff(fact_7305_of__int__divide__in__Ints,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A)
     => member_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(ring_1_of_int_int,A)),aa_int_int(ring_1_of_int_int,B2)),ring_1_Ints_int) ) ).

% of_int_divide_in_Ints
tff(fact_7306_inverse__le__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ) ).

% inverse_le_iff
tff(fact_7307_inverse__le__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ) ).

% inverse_le_iff
tff(fact_7308_inverse__less__iff,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2))
    <=> ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2))
         => aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) )
        & ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)),zero_zero_real)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ) ).

% inverse_less_iff
tff(fact_7309_inverse__less__iff,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2))
    <=> ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2))
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) )
        & ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)),zero_zero_rat)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ) ).

% inverse_less_iff
tff(fact_7310_one__le__inverse,axiom,
    ! [A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),one_one_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,A)) ) ) ).

% one_le_inverse
tff(fact_7311_one__le__inverse,axiom,
    ! [A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),one_one_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,A)) ) ) ).

% one_le_inverse
tff(fact_7312_inverse__less__1__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,Xb)),one_one_real)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_7313_inverse__less__1__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,Xb)),one_one_rat)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),zero_zero_rat)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),Xb) ) ) ).

% inverse_less_1_iff
tff(fact_7314_one__le__inverse__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(inverse_inverse_real,Xb))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% one_le_inverse_iff
tff(fact_7315_one__le__inverse__iff,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(inverse_inverse_rat,Xb))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),one_one_rat) ) ) ).

% one_le_inverse_iff
tff(fact_7316_inverse__diff__inverse,axiom,
    ! [A: real,B2: real] :
      ( ( A != zero_zero_real )
     => ( ( B2 != zero_zero_real )
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(inverse_inverse_real,B2)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,A)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2))),aa_real_real(inverse_inverse_real,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7317_inverse__diff__inverse,axiom,
    ! [A: complex,B2: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B2 != zero_zero_complex )
       => ( aa_complex_complex(minus_minus_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(invers8013647133539491842omplex,B2)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(aa_complex_complex(invers8013647133539491842omplex,A)),aa_complex_complex(minus_minus_complex(A),B2))),aa_complex_complex(invers8013647133539491842omplex,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7318_inverse__diff__inverse,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B2 != zero_zero_rat )
       => ( aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(inverse_inverse_rat,B2)) = aa_rat_rat(uminus_uminus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(inverse_inverse_rat,A)),aa_rat_rat(minus_minus_rat(A),B2))),aa_rat_rat(inverse_inverse_rat,B2))) ) ) ) ).

% inverse_diff_inverse
tff(fact_7319_reals__Archimedean,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2)))),Xb) ) ).

% reals_Archimedean
tff(fact_7320_reals__Archimedean,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,N2)))),Xb) ) ).

% reals_Archimedean
tff(fact_7321_gbinomial__addition__formula,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_addition_formula
tff(fact_7322_gbinomial__addition__formula,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_addition_formula
tff(fact_7323_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_addition_formula
tff(fact_7324_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorb_comp
tff(fact_7325_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorb_comp
tff(fact_7326_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorb_comp
tff(fact_7327_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7328_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(power_power_rat(aa_rat_rat(divide_divide_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),K)),aa_nat_rat(gbinomial_rat(A),K)) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_7329_gbinomial__mult__1_H,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),A) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7330_gbinomial__mult__1_H,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),A) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7331_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1'
tff(fact_7332_gbinomial__mult__1,axiom,
    ! [A: complex,K: nat] : aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(A),K)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),aa_nat_complex(gbinomial_complex(A),K))),aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7333_gbinomial__mult__1,axiom,
    ! [A: real,K: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7334_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(A),K)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,K)),aa_nat_rat(gbinomial_rat(A),K))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)))) ).

% gbinomial_mult_1
tff(fact_7335_forall__pos__mono__1,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N2: nat] : aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,N2))))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono_1
tff(fact_7336_forall__pos__mono,axiom,
    ! [P: fun_real_o,E: real] :
      ( ! [D2: real,E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,D2),E2)
         => ( aa_real_o(P,D2)
           => aa_real_o(P,E2) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero_nat )
           => aa_real_o(P,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))) )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
         => aa_real_o(P,E) ) ) ) ).

% forall_pos_mono
tff(fact_7337_real__arch__inverse,axiom,
    ! [E: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
    <=> ? [N4: nat] :
          ( ( N4 != zero_zero_nat )
          & aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4)))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N4))),E) ) ) ).

% real_arch_inverse
tff(fact_7338_even__xor__iff,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se3222712562003087583nteger(A,B2))
    <=> ( 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))),B2) ) ) ).

% even_xor_iff
tff(fact_7339_even__xor__iff,axiom,
    ! [A: nat,B2: 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_se6528837805403552850or_nat,A),B2))
    <=> ( 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))),B2) ) ) ).

% even_xor_iff
tff(fact_7340_even__xor__iff,axiom,
    ! [A: int,B2: 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_se6526347334894502574or_int(A),B2))
    <=> ( 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))),B2) ) ) ).

% even_xor_iff
tff(fact_7341_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sqrt,Xb)),Xb) = aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_7342_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(ln_ln_real,Xb)) ) ) ).

% ln_inverse
tff(fact_7343_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( member_real(A,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),A)),A)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,A),zero_zero_real) ) ) ).

% Ints_odd_less_0
tff(fact_7344_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( member_rat(A,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),A)),A)),zero_zero_rat)
      <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),zero_zero_rat) ) ) ).

% Ints_odd_less_0
tff(fact_7345_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( member_int(A,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),A)),A)),zero_zero_int)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Ints_odd_less_0
tff(fact_7346_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( ( Xb != zero_z3403309356797280102nteger )
       => aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,one_one_Code_integer),abs_abs_Code_integer(Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7347_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( ( Xb != zero_zero_real )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(abs_abs_real,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7348_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( ( Xb != zero_zero_rat )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_rat_rat(abs_abs_rat,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7349_Ints__nonzero__abs__ge1,axiom,
    ! [Xb: int] :
      ( member_int(Xb,ring_1_Ints_int)
     => ( ( Xb != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(abs_abs_int,Xb)) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_7350_Ints__nonzero__abs__less1,axiom,
    ! [Xb: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(Xb)),one_one_Code_integer)
       => ( Xb = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7351_Ints__nonzero__abs__less1,axiom,
    ! [Xb: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
       => ( Xb = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7352_Ints__nonzero__abs__less1,axiom,
    ! [Xb: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,Xb)),one_one_rat)
       => ( Xb = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7353_Ints__nonzero__abs__less1,axiom,
    ! [Xb: int] :
      ( member_int(Xb,ring_1_Ints_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,Xb)),one_one_int)
       => ( Xb = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_7354_Ints__eq__abs__less1,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( member_Code_integer(Xb,ring_11222124179247155820nteger)
     => ( member_Code_integer(Y,ring_11222124179247155820nteger)
       => ( ( Xb = Y )
        <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(Xb,Y))),one_one_Code_integer) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7355_Ints__eq__abs__less1,axiom,
    ! [Xb: real,Y: real] :
      ( member_real(Xb,ring_1_Ints_real)
     => ( member_real(Y,ring_1_Ints_real)
       => ( ( Xb = Y )
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y))),one_one_real) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7356_Ints__eq__abs__less1,axiom,
    ! [Xb: rat,Y: rat] :
      ( member_rat(Xb,ring_1_Ints_rat)
     => ( member_rat(Y,ring_1_Ints_rat)
       => ( ( Xb = Y )
        <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(Xb),Y))),one_one_rat) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7357_Ints__eq__abs__less1,axiom,
    ! [Xb: int,Y: int] :
      ( member_int(Xb,ring_1_Ints_int)
     => ( member_int(Y,ring_1_Ints_int)
       => ( ( Xb = Y )
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb),Y))),one_one_int) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_7358_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),pi)) = zero_zero_real )
    <=> member_real(Xb,ring_1_Ints_real) ) ).

% sin_times_pi_eq_0
tff(fact_7359_ex__inverse__of__nat__less,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,N2))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7360_ex__inverse__of__nat__less,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),Xb)
     => ? [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N2)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(inverse_inverse_rat,aa_nat_rat(semiri681578069525770553at_rat,N2))),Xb) ) ) ).

% ex_inverse_of_nat_less
tff(fact_7361_power__diff__conv__inverse,axiom,
    ! [Xb: real,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7362_power__diff__conv__inverse,axiom,
    ! [Xb: complex,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_complex(power_power_complex(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(Xb),Nb)),aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7363_power__diff__conv__inverse,axiom,
    ! [Xb: rat,Mb: nat,Nb: nat] :
      ( ( Xb != zero_zero_rat )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
       => ( aa_nat_rat(power_power_rat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(Xb),Nb)),aa_nat_rat(power_power_rat(aa_rat_rat(inverse_inverse_rat,Xb)),Mb)) ) ) ) ).

% power_diff_conv_inverse
tff(fact_7364_Suc__times__gbinomial,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(gbinomial_complex(A),K)) ).

% Suc_times_gbinomial
tff(fact_7365_Suc__times__gbinomial,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% Suc_times_gbinomial
tff(fact_7366_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(gbinomial_rat(A),K)) ).

% Suc_times_gbinomial
tff(fact_7367_gbinomial__absorption,axiom,
    ! [K: nat,A: complex] : aa_complex_complex(times_times_complex(aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K))),aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K))) = aa_complex_complex(times_times_complex(A),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ).

% gbinomial_absorption
tff(fact_7368_gbinomial__absorption,axiom,
    ! [K: nat,A: real] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ).

% gbinomial_absorption
tff(fact_7369_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K))),aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ).

% gbinomial_absorption
tff(fact_7370_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),Mb)),aa_nat_complex(gbinomial_complex(aa_nat_complex(semiri8010041392384452111omplex,Mb)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7371_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),Mb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_nat_real(semiri5074537144036343181t_real,Mb)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7372_gbinomial__trinomial__revision,axiom,
    ! [K: nat,Mb: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),Mb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),Mb)),aa_nat_rat(gbinomial_rat(aa_nat_rat(semiri681578069525770553at_rat,Mb)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),K))) ) ) ).

% gbinomial_trinomial_revision
tff(fact_7373_log__inverse,axiom,
    ! [A: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),A)
     => ( ( A != one_one_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_real(aa_rea7340605660300375871l_real(log,A),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(log,A),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_7374_frac__neg,axiom,
    ! [Xb: real] :
      aa_real_real(archim2898591450579166408c_real,aa_real_real(uminus_uminus_real,Xb)) = $ite(member_real(Xb,ring_1_Ints_real),zero_zero_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_real_real(archim2898591450579166408c_real,Xb))) ).

% frac_neg
tff(fact_7375_frac__neg,axiom,
    ! [Xb: rat] :
      aa_rat_rat(archimedean_frac_rat,aa_rat_rat(uminus_uminus_rat,Xb)) = $ite(member_rat(Xb,ring_1_Ints_rat),zero_zero_rat,aa_rat_rat(minus_minus_rat(one_one_rat),aa_rat_rat(archimedean_frac_rat,Xb))) ).

% frac_neg
tff(fact_7376_gbinomial__factors,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))),aa_nat_complex(gbinomial_complex(A),K)) ).

% gbinomial_factors
tff(fact_7377_gbinomial__factors,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)) ).

% gbinomial_factors
tff(fact_7378_gbinomial__factors,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))),aa_nat_rat(gbinomial_rat(A),K)) ).

% gbinomial_factors
tff(fact_7379_gbinomial__rec,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_nat(suc,K)) = aa_complex_complex(times_times_complex(aa_nat_complex(gbinomial_complex(A),K)),aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(A),one_one_complex)),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7380_gbinomial__rec,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7381_gbinomial__rec,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_nat(suc,K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(gbinomial_rat(A),K)),aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(suc,K)))) ).

% gbinomial_rec
tff(fact_7382_gbinomial__negated__upper,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(minus_minus_complex(aa_nat_complex(semiri8010041392384452111omplex,K)),A)),one_one_complex)),K)) ).

% gbinomial_negated_upper
tff(fact_7383_gbinomial__negated__upper,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(semiri5074537144036343181t_real,K)),A)),one_one_real)),K)) ).

% gbinomial_negated_upper
tff(fact_7384_gbinomial__negated__upper,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(minus_minus_rat(aa_nat_rat(semiri681578069525770553at_rat,K)),A)),one_one_rat)),K)) ).

% gbinomial_negated_upper
tff(fact_7385_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,Nb))),one_one_complex)),K)) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),Nb)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(uminus1482373934393186551omplex,aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),Nb)) ).

% gbinomial_index_swap
tff(fact_7386_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),one_one_real)),K)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Nb)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),Nb)) ).

% gbinomial_index_swap
tff(fact_7387_gbinomial__index__swap,axiom,
    ! [K: nat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,Nb))),one_one_rat)),K)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),Nb)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(uminus_uminus_rat,aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),Nb)) ).

% gbinomial_index_swap
tff(fact_7388_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_7389_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7390_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_rat(ring_1_of_int_rat,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7391_le__mult__floor__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim6058952711729229775r_real,A)),aa_real_int(archim6058952711729229775r_real,B2)))),aa_int_int(ring_1_of_int_int,aa_real_int(archim6058952711729229775r_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7392_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_real(ring_1_of_int_real,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7393_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7394_le__mult__floor__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim3151403230148437115or_rat,A)),aa_rat_int(archim3151403230148437115or_rat,B2)))),aa_int_int(ring_1_of_int_int,aa_rat_int(archim3151403230148437115or_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))) ) ) ).

% le_mult_floor_Ints
tff(fact_7395_frac__unique__iff,axiom,
    ! [Xb: real,A: real] :
      ( ( aa_real_real(archim2898591450579166408c_real,Xb) = A )
    <=> ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),A),ring_1_Ints_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,A),one_one_real) ) ) ).

% frac_unique_iff
tff(fact_7396_frac__unique__iff,axiom,
    ! [Xb: rat,A: rat] :
      ( ( aa_rat_rat(archimedean_frac_rat,Xb) = A )
    <=> ( member_rat(aa_rat_rat(minus_minus_rat(Xb),A),ring_1_Ints_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,A),one_one_rat) ) ) ).

% frac_unique_iff
tff(fact_7397_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7398_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7399_mult__ceiling__le__Ints,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),A)
     => ( member_real(A,ring_1_Ints_real)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_real_int(archim7802044766580827645g_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),B2)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_real_int(archim7802044766580827645g_real,A)),aa_real_int(archim7802044766580827645g_real,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7400_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7401_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7402_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),A)
     => ( member_rat(A,ring_1_Ints_rat)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(ring_1_of_int_int,aa_rat_int(archim2889992004027027881ng_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B2)))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_rat_int(archim2889992004027027881ng_rat,A)),aa_rat_int(archim2889992004027027881ng_rat,B2)))) ) ) ).

% mult_ceiling_le_Ints
tff(fact_7403_gbinomial__minus,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(aa_complex_complex(uminus1482373934393186551omplex,A)),K) = aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(aa_complex_complex(plus_plus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)) ).

% gbinomial_minus
tff(fact_7404_gbinomial__minus,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(uminus_uminus_real,A)),K) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)) ).

% gbinomial_minus
tff(fact_7405_gbinomial__minus,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(aa_rat_rat(uminus_uminus_rat,A)),K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)) ).

% gbinomial_minus
tff(fact_7406_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))) ) ).

% plus_inverse_ge_2
tff(fact_7407_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(inverse_inverse_real,Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_7408_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(plus_plus_complex(aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_complex(gbinomial_complex(aa_complex_complex(minus_minus_complex(A),one_one_complex)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7409_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_real(aa_real_fun_nat_real(gbinomial_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),one_one_real)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7410_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),K)
     => ( aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat))),aa_nat_rat(gbinomial_rat(aa_rat_rat(minus_minus_rat(A),one_one_rat)),K)) ) ) ).

% gbinomial_reduce_nat
tff(fact_7411_gbinomial__pochhammer,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(aa_nat_complex(power_power_complex(aa_complex_complex(uminus1482373934393186551omplex,one_one_complex)),K)),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(uminus1482373934393186551omplex,A)),K))),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer
tff(fact_7412_gbinomial__pochhammer,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(power_power_rat(aa_rat_rat(uminus_uminus_rat,one_one_rat)),K)),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(uminus_uminus_rat,A)),K))),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer
tff(fact_7413_gbinomial__pochhammer,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),K)),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(uminus_uminus_real,A)),K))),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer
tff(fact_7414_gbinomial__pochhammer_H,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),K) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(minus_minus_complex(A),aa_nat_complex(semiri8010041392384452111omplex,K))),one_one_complex)),K)),aa_nat_complex(semiri5044797733671781792omplex,K)) ).

% gbinomial_pochhammer'
tff(fact_7415_gbinomial__pochhammer_H,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),K) = aa_rat_rat(divide_divide_rat(aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(minus_minus_rat(A),aa_nat_rat(semiri681578069525770553at_rat,K))),one_one_rat)),K)),aa_nat_rat(semiri773545260158071498ct_rat,K)) ).

% gbinomial_pochhammer'
tff(fact_7416_gbinomial__pochhammer_H,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),aa_nat_real(semiri5074537144036343181t_real,K))),one_one_real)),K)),aa_nat_real(semiri2265585572941072030t_real,K)) ).

% gbinomial_pochhammer'
tff(fact_7417_tan__cot,axiom,
    ! [Xb: real] : aa_real_real(tan_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))),Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(tan_real,Xb)) ).

% tan_cot
tff(fact_7418_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = zero_zero_real ) ) ).

% sin_integer_2pi
tff(fact_7419_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member_real(Nb,ring_1_Ints_real)
     => ( aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),Nb)) = one_one_real ) ) ).

% cos_integer_2pi
tff(fact_7420_xor__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% xor_nat_unfold
tff(fact_7421_xor__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) != ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% xor_nat_rec
tff(fact_7422_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ) ).

% real_le_x_sinh
tff(fact_7423_xor__one__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(A,one_one_Code_integer) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% xor_one_eq
tff(fact_7424_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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)))),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))) ).

% xor_one_eq
tff(fact_7425_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_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)))),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))) ).

% xor_one_eq
tff(fact_7426_one__xor__eq,axiom,
    ! [A: code_integer] : bit_se3222712562003087583nteger(one_one_Code_integer,A) = minus_8373710615458151222nteger(plus_p5714425477246183910nteger(A,zero_n356916108424825756nteger(aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A))) ).

% one_xor_eq
tff(fact_7427_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),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)))),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_xor_eq
tff(fact_7428_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(bit_se6526347334894502574or_int(one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_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)))),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_xor_eq
tff(fact_7429_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Xb)),aa_real_real(inverse_inverse_real,aa_real_real(exp_real,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% real_le_abs_sinh
tff(fact_7430_tan__sec,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cos_real,Xb) != zero_zero_real )
     => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(tan_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(inverse_inverse_real,aa_real_real(cos_real,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7431_tan__sec,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cos_complex,Xb) != zero_zero_complex )
     => ( aa_complex_complex(plus_plus_complex(one_one_complex),aa_nat_complex(power_power_complex(aa_complex_complex(tan_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_complex(power_power_complex(aa_complex_complex(invers8013647133539491842omplex,aa_complex_complex(cos_complex,Xb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ) ) ).

% tan_sec
tff(fact_7432_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(sinh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% sinh_ln_real
tff(fact_7433_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_real(cosh_real,aa_real_real(ln_ln_real,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),aa_real_real(inverse_inverse_real,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ).

% cosh_ln_real
tff(fact_7434_cosh__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa_real_real(cosh_real,Xb) = zero_zero_real )
    <=> ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(exp_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(uminus_uminus_real,one_one_real) ) ) ).

% cosh_zero_iff
tff(fact_7435_cosh__zero__iff,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) = zero_zero_complex )
    <=> ( aa_nat_complex(power_power_complex(aa_complex_complex(exp_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(uminus1482373934393186551omplex,one_one_complex) ) ) ).

% cosh_zero_iff
tff(fact_7436_psubsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_eq_set_nat(A3,B3)
     => ( ( A3 != B3 )
       => ord_less_set_nat(A3,B3) ) ) ).

% psubsetI
tff(fact_7437_push__bit__numeral__minus__1,axiom,
    ! [Nb: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% push_bit_numeral_minus_1
tff(fact_7438_push__bit__numeral__minus__1,axiom,
    ! [Nb: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,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))),aa_num_nat(numeral_numeral_nat,Nb))) ).

% push_bit_numeral_minus_1
tff(fact_7439_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_7440_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_7441_push__bit__of__0,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),zero_zero_int) = zero_zero_int ).

% push_bit_of_0
tff(fact_7442_push__bit__of__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),zero_zero_nat) = zero_zero_nat ).

% push_bit_of_0
tff(fact_7443_push__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: int] :
      ( ( aa_int_int(bit_se545348938243370406it_int(Nb),A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
tff(fact_7444_push__bit__eq__0__iff,axiom,
    ! [Nb: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A) = zero_zero_nat )
    <=> ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
tff(fact_7445_push__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Mb),aa_int_int(bit_se545348938243370406it_int(Nb),A)) = aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% push_bit_push_bit
tff(fact_7446_push__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),A) ).

% push_bit_push_bit
tff(fact_7447_push__bit__and,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(bit_se725231765392027082nd_int(A),B2)) = aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_se545348938243370406it_int(Nb),A)),aa_int_int(bit_se545348938243370406it_int(Nb),B2)) ).

% push_bit_and
tff(fact_7448_push__bit__and,axiom,
    ! [Nb: nat,A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),B2)) ).

% push_bit_and
tff(fact_7449_push__bit__xor,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(bit_se6526347334894502574or_int(A),B2)) = aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_se545348938243370406it_int(Nb),A)),aa_int_int(bit_se545348938243370406it_int(Nb),B2)) ).

% push_bit_xor
tff(fact_7450_push__bit__xor,axiom,
    ! [Nb: nat,A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),B2)) ).

% push_bit_xor
tff(fact_7451_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_7452_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_7453_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_7454_cosh__0,axiom,
    aa_complex_complex(cosh_complex,zero_zero_complex) = one_one_complex ).

% cosh_0
tff(fact_7455_cosh__0,axiom,
    aa_real_real(cosh_real,zero_zero_real) = one_one_real ).

% cosh_0
tff(fact_7456_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb) ) ).

% sinh_real_pos_iff
tff(fact_7457_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ).

% sinh_real_neg_iff
tff(fact_7458_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(sinh_real,Xb))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_7459_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),zero_zero_real)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ).

% sinh_real_nonpos_iff
tff(fact_7460_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_7461_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_7462_push__bit__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),aa_num_int(numeral_numeral_int,K)) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_7463_push__bit__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Nb)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% push_bit_Suc_numeral
tff(fact_7464_push__bit__Suc__minus__numeral,axiom,
    ! [Nb: nat,K: num] : bit_se7788150548672797655nteger(aa_nat_nat(suc,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = bit_se7788150548672797655nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_7465_push__bit__Suc__minus__numeral,axiom,
    ! [Nb: nat,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_Suc_minus_numeral
tff(fact_7466_push__bit__numeral,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,L)),aa_num_int(numeral_numeral_int,K)) = aa_int_int(bit_se545348938243370406it_int(aa_num_nat(pred_numeral,L)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,K))) ).

% push_bit_numeral
tff(fact_7467_push__bit__numeral,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_num_nat(numeral_numeral_nat,L)),aa_num_nat(numeral_numeral_nat,K)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_num_nat(pred_numeral,L)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,K))) ).

% push_bit_numeral
tff(fact_7468_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_7469_push__bit__Suc,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),A) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_7470_push__bit__Suc,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,Nb)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% push_bit_Suc
tff(fact_7471_push__bit__of__1,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int) = aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb) ).

% push_bit_of_1
tff(fact_7472_push__bit__of__1,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),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) ).

% push_bit_of_1
tff(fact_7473_even__push__bit__iff,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_se7788150548672797655nteger(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_push_bit_iff
tff(fact_7474_even__push__bit__iff,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_se545348938243370406it_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_push_bit_iff
tff(fact_7475_even__push__bit__iff,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_se547839408752420682it_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_push_bit_iff
tff(fact_7476_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : bit_se7788150548672797655nteger(aa_num_nat(numeral_numeral_nat,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(K))) = bit_se7788150548672797655nteger(aa_num_nat(pred_numeral,L),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_7477_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se545348938243370406it_int(aa_num_nat(numeral_numeral_nat,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_se545348938243370406it_int(aa_num_nat(pred_numeral,L)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,K)))) ).

% push_bit_minus_numeral
tff(fact_7478_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_7479_push__bit__add,axiom,
    ! [Nb: nat,A: int,B2: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se545348938243370406it_int(Nb),A)),aa_int_int(bit_se545348938243370406it_int(Nb),B2)) ).

% push_bit_add
tff(fact_7480_push__bit__add,axiom,
    ! [Nb: nat,A: nat,B2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),B2)) ).

% push_bit_add
tff(fact_7481_cosh__plus__sinh,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(sinh_complex,Xb)) = aa_complex_complex(exp_complex,Xb) ).

% cosh_plus_sinh
tff(fact_7482_cosh__plus__sinh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% cosh_plus_sinh
tff(fact_7483_sinh__plus__cosh,axiom,
    ! [Xb: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) = aa_complex_complex(exp_complex,Xb) ).

% sinh_plus_cosh
tff(fact_7484_sinh__plus__cosh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) = aa_real_real(exp_real,Xb) ).

% sinh_plus_cosh
tff(fact_7485_psubsetD,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat,C: extended_enat] :
      ( ord_le2529575680413868914d_enat(A3,B3)
     => ( member_Extended_enat(C,A3)
       => member_Extended_enat(C,B3) ) ) ).

% psubsetD
tff(fact_7486_psubsetD,axiom,
    ! [A3: set_complex,B3: set_complex,C: complex] :
      ( ord_less_set_complex(A3,B3)
     => ( member_complex(C,A3)
       => member_complex(C,B3) ) ) ).

% psubsetD
tff(fact_7487_psubsetD,axiom,
    ! [A3: set_real,B3: set_real,C: real] :
      ( ord_less_set_real(A3,B3)
     => ( member_real(C,A3)
       => member_real(C,B3) ) ) ).

% psubsetD
tff(fact_7488_psubsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C: nat] :
      ( ord_less_set_nat(A3,B3)
     => ( member_nat(C,A3)
       => member_nat(C,B3) ) ) ).

% psubsetD
tff(fact_7489_psubsetD,axiom,
    ! [A3: set_int,B3: set_int,C: int] :
      ( ord_less_set_int(A3,B3)
     => ( member_int(C,A3)
       => member_int(C,B3) ) ) ).

% psubsetD
tff(fact_7490_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_le_cosh_real
tff(fact_7491_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% sinh_less_cosh_real
tff(fact_7492_push__bit__of__int,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(ring_1_of_int_int,K)) = aa_int_int(ring_1_of_int_int,aa_int_int(bit_se545348938243370406it_int(Nb),K)) ).

% push_bit_of_int
tff(fact_7493_of__nat__push__bit,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Nb)) = aa_int_int(bit_se545348938243370406it_int(Mb),aa_nat_int(semiri1314217659103216013at_int,Nb)) ).

% of_nat_push_bit
tff(fact_7494_of__nat__push__bit,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),aa_nat_nat(semiri1316708129612266289at_nat,Nb)) ).

% of_nat_push_bit
tff(fact_7495_push__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_nat_int(semiri1314217659103216013at_int,Mb)) = aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb)) ).

% push_bit_of_nat
tff(fact_7496_push__bit__of__nat,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(semiri1316708129612266289at_nat,Mb)) = aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb)) ).

% push_bit_of_nat
tff(fact_7497_psubset__imp__ex__mem,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat] :
      ( ord_le2529575680413868914d_enat(A3,B3)
     => ? [B4: extended_enat] : member_Extended_enat(B4,minus_925952699566721837d_enat(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_7498_psubset__imp__ex__mem,axiom,
    ! [A3: set_complex,B3: set_complex] :
      ( ord_less_set_complex(A3,B3)
     => ? [B4: complex] : member_complex(B4,minus_811609699411566653omplex(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_7499_psubset__imp__ex__mem,axiom,
    ! [A3: set_real,B3: set_real] :
      ( ord_less_set_real(A3,B3)
     => ? [B4: real] : member_real(B4,minus_minus_set_real(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_7500_psubset__imp__ex__mem,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ord_less_set_int(A3,B3)
     => ? [B4: int] : member_int(B4,minus_minus_set_int(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_7501_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
     => ? [B4: nat] : member_nat(B4,minus_minus_set_nat(B3,A3)) ) ).

% psubset_imp_ex_mem
tff(fact_7502_sinh__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% sinh_diff
tff(fact_7503_cosh__diff,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% cosh_diff
tff(fact_7504_sinh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% sinh_add
tff(fact_7505_cosh__add,axiom,
    ! [Xb: real,Y: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sinh_real,Xb)),aa_real_real(sinh_real,Y))) ).

% cosh_add
tff(fact_7506_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_7507_push__bit__minus,axiom,
    ! [Nb: nat,A: code_integer] : bit_se7788150548672797655nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_se7788150548672797655nteger(Nb,A)) ).

% push_bit_minus
tff(fact_7508_push__bit__minus,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,A)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se545348938243370406it_int(Nb),A)) ).

% push_bit_minus
tff(fact_7509_tanh__def,axiom,
    ! [Xb: complex] : aa_complex_complex(tanh_complex,Xb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) ).

% tanh_def
tff(fact_7510_tanh__def,axiom,
    ! [Xb: real] : aa_real_real(tanh_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) ).

% tanh_def
tff(fact_7511_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_7512_sinh__minus__cosh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sinh_real,Xb)),aa_real_real(cosh_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb))) ).

% sinh_minus_cosh
tff(fact_7513_sinh__minus__cosh,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(sinh_complex,Xb)),aa_complex_complex(cosh_complex,Xb)) = aa_complex_complex(uminus1482373934393186551omplex,aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb))) ).

% sinh_minus_cosh
tff(fact_7514_cosh__minus__sinh,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(cosh_real,Xb)),aa_real_real(sinh_real,Xb)) = aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Xb)) ).

% cosh_minus_sinh
tff(fact_7515_cosh__minus__sinh,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_complex_complex(cosh_complex,Xb)),aa_complex_complex(sinh_complex,Xb)) = aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Xb)) ).

% cosh_minus_sinh
tff(fact_7516_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se6526347334894502574or_int(Xb),Y)) ) ) ).

% XOR_lower
tff(fact_7517_cosh__real__pos,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_pos
tff(fact_7518_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_nonneg
tff(fact_7519_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_7520_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_7521_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(cosh_real,Xb)) ).

% cosh_real_ge_1
tff(fact_7522_sinh__double,axiom,
    ! [Xb: complex] : aa_complex_complex(sinh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_complex_complex(sinh_complex,Xb))),aa_complex_complex(cosh_complex,Xb)) ).

% sinh_double
tff(fact_7523_sinh__double,axiom,
    ! [Xb: real] : aa_real_real(sinh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(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(sinh_real,Xb))),aa_real_real(cosh_real,Xb)) ).

% sinh_double
tff(fact_7524_push__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Mb),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) = aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_int_int(bit_se545348938243370406it_int(Mb),A)) ).

% push_bit_take_bit
tff(fact_7525_push__bit__take__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),A)) ).

% push_bit_take_bit
tff(fact_7526_take__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_se545348938243370406it_int(Nb),A)) = aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),A)) ).

% take_bit_push_bit
tff(fact_7527_take__bit__push__bit,axiom,
    ! [Mb: nat,Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),A)) ).

% take_bit_push_bit
tff(fact_7528_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),Y) = aa_complex_complex(times_times_complex(Xb),aa_complex_complex(invers8013647133539491842omplex,Y)) ).

% divide_complex_def
tff(fact_7529_flip__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_7530_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_7531_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_7532_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),zero_zero_real)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(cosh_real,Xb)),aa_real_real(cosh_real,Y))
        <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_7533_bit__push__bit__iff__int,axiom,
    ! [Mb: nat,K: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_se545348938243370406it_int(Mb),K)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(bit_se1146084159140164899it_int(K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_int
tff(fact_7534_cosh__square__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% cosh_square_eq
tff(fact_7535_cosh__square__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% cosh_square_eq
tff(fact_7536_sinh__square__eq,axiom,
    ! [Xb: complex] : aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_complex) ).

% sinh_square_eq
tff(fact_7537_sinh__square__eq,axiom,
    ! [Xb: real] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real) ).

% sinh_square_eq
tff(fact_7538_hyperbolic__pythagoras,axiom,
    ! [Xb: complex] : aa_complex_complex(minus_minus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_complex ).

% hyperbolic_pythagoras
tff(fact_7539_hyperbolic__pythagoras,axiom,
    ! [Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ).

% hyperbolic_pythagoras
tff(fact_7540_xor__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se6526347334894502574or_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% xor_nat_def
tff(fact_7541_psubsetE,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
     => ~ ( ord_less_eq_set_nat(A3,B3)
         => ord_less_eq_set_nat(B3,A3) ) ) ).

% psubsetE
tff(fact_7542_psubset__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
    <=> ( ord_less_eq_set_nat(A3,B3)
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_7543_psubset__imp__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
     => ord_less_eq_set_nat(A3,B3) ) ).

% psubset_imp_subset
tff(fact_7544_psubset__subset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C5: set_nat] :
      ( ord_less_set_nat(A3,B3)
     => ( ord_less_eq_set_nat(B3,C5)
       => ord_less_set_nat(A3,C5) ) ) ).

% psubset_subset_trans
tff(fact_7545_subset__not__subset__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
    <=> ( ord_less_eq_set_nat(A3,B3)
        & ~ ord_less_eq_set_nat(B3,A3) ) ) ).

% subset_not_subset_eq
tff(fact_7546_subset__psubset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C5: set_nat] :
      ( ord_less_eq_set_nat(A3,B3)
     => ( ord_less_set_nat(B3,C5)
       => ord_less_set_nat(A3,C5) ) ) ).

% subset_psubset_trans
tff(fact_7547_subset__iff__psubset__eq,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_eq_set_nat(A3,B3)
    <=> ( ord_less_set_nat(A3,B3)
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_7548_double__diff,axiom,
    ! [A3: set_nat,B3: set_nat,C5: set_nat] :
      ( ord_less_eq_set_nat(A3,B3)
     => ( ord_less_eq_set_nat(B3,C5)
       => ( minus_minus_set_nat(B3,minus_minus_set_nat(C5,A3)) = A3 ) ) ) ).

% double_diff
tff(fact_7549_Diff__subset,axiom,
    ! [A3: set_nat,B3: set_nat] : ord_less_eq_set_nat(minus_minus_set_nat(A3,B3),A3) ).

% Diff_subset
tff(fact_7550_Diff__mono,axiom,
    ! [A3: set_nat,C5: set_nat,D4: set_nat,B3: set_nat] :
      ( ord_less_eq_set_nat(A3,C5)
     => ( ord_less_eq_set_nat(D4,B3)
       => ord_less_eq_set_nat(minus_minus_set_nat(A3,B3),minus_minus_set_nat(C5,D4)) ) ) ).

% Diff_mono
tff(fact_7551_bit__push__bit__iff__nat,axiom,
    ! [Mb: 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,Mb),Q2)),Nb)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
        & aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,Q2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_7552_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : aa_int_int(bit_concat_bit(Nb,K),L) = aa_int_int(aa_int_fun_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_7553_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_real(arcosh_real,aa_real_real(cosh_real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_7554_flip__bit__eq__xor,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(Nb),A) = aa_int_int(bit_se6526347334894502574or_int(A),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) ).

% flip_bit_eq_xor
tff(fact_7555_flip__bit__eq__xor,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,Nb),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),one_one_nat)) ).

% flip_bit_eq_xor
tff(fact_7556_cosh__double,axiom,
    ! [Xb: complex] : aa_complex_complex(cosh_complex,aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),Xb)) = aa_complex_complex(plus_plus_complex(aa_nat_complex(power_power_complex(aa_complex_complex(cosh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_complex(power_power_complex(aa_complex_complex(sinh_complex,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7557_cosh__double,axiom,
    ! [Xb: real] : aa_real_real(cosh_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(cosh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(sinh_real,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cosh_double
tff(fact_7558_push__bit__double,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se545348938243370406it_int(Nb),A)),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% push_bit_double
tff(fact_7559_push__bit__double,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(times_times_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_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),A)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% push_bit_double
tff(fact_7560_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(A),Nb)
    <=> ( aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int)) != zero_zero_int ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7561_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),Nb)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),one_one_nat)) != zero_zero_nat ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_7562_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se545348938243370406it_int(Nb),K) = aa_int_int(aa_int_fun_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_7563_push__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_nat_def
tff(fact_7564_push__bit__eq__mult,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(Nb),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ).

% push_bit_eq_mult
tff(fact_7565_push__bit__eq__mult,axiom,
    ! [Nb: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,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,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% push_bit_eq_mult
tff(fact_7566_exp__dvdE,axiom,
    ! [Nb: nat,A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)),A)
     => ~ ! [B4: code_integer] : A != bit_se7788150548672797655nteger(Nb,B4) ) ).

% exp_dvdE
tff(fact_7567_exp__dvdE,axiom,
    ! [Nb: nat,A: 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)
     => ~ ! [B4: int] : A != aa_int_int(bit_se545348938243370406it_int(Nb),B4) ) ).

% exp_dvdE
tff(fact_7568_exp__dvdE,axiom,
    ! [Nb: nat,A: 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)
     => ~ ! [B4: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Nb),B4) ) ).

% exp_dvdE
tff(fact_7569_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_7570_tanh__add,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_complex(cosh_complex,Xb) != zero_zero_complex )
     => ( ( aa_complex_complex(cosh_complex,Y) != zero_zero_complex )
       => ( aa_complex_complex(tanh_complex,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y))),aa_complex_complex(plus_plus_complex(one_one_complex),aa_complex_complex(times_times_complex(aa_complex_complex(tanh_complex,Xb)),aa_complex_complex(tanh_complex,Y)))) ) ) ) ).

% tanh_add
tff(fact_7571_tanh__add,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa_real_real(cosh_real,Xb) != zero_zero_real )
     => ( ( aa_real_real(cosh_real,Y) != zero_zero_real )
       => ( aa_real_real(tanh_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(tanh_real,Xb)),aa_real_real(tanh_real,Y)))) ) ) ) ).

% tanh_add
tff(fact_7572_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se6526347334894502574or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_7573_cosh__field__def,axiom,
    ! [Zb: real] : aa_real_real(cosh_real,Zb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(exp_real,Zb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_7574_cosh__field__def,axiom,
    ! [Zb: complex] : aa_complex_complex(cosh_complex,Zb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(plus_plus_complex(aa_complex_complex(exp_complex,Zb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% cosh_field_def
tff(fact_7575_complex__inverse,axiom,
    ! [A: real,B2: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(complex2(A),B2)) = 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,B2),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,B2)),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,B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% complex_inverse
tff(fact_7576_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se6526347334894502574or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K) != ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% xor_int_rec
tff(fact_7577_sinh__field__def,axiom,
    ! [Zb: real] : aa_real_real(sinh_real,Zb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(exp_real,Zb)),aa_real_real(exp_real,aa_real_real(uminus_uminus_real,Zb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7578_sinh__field__def,axiom,
    ! [Zb: complex] : aa_complex_complex(sinh_complex,Zb) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(minus_minus_complex(aa_complex_complex(exp_complex,Zb)),aa_complex_complex(exp_complex,aa_complex_complex(uminus1482373934393186551omplex,Zb)))),numera6690914467698888265omplex(aa_num_num(bit0,one))) ).

% sinh_field_def
tff(fact_7579_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se6526347334894502574or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_7580_True,axiom,
    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,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList)) ).

% True
tff(fact_7581_Cauchy__iff2,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [J3: nat] :
        ? [M7: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M6)
         => ! [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(X5,M6)),aa_nat_real(X5,N4)))),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_7582_dual__order_Orefl,axiom,
    ! [A: set_nat] : ord_less_eq_set_nat(A,A) ).

% dual_order.refl
tff(fact_7583_dual__order_Orefl,axiom,
    ! [A: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),A) ).

% dual_order.refl
tff(fact_7584_dual__order_Orefl,axiom,
    ! [A: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),A) ).

% dual_order.refl
tff(fact_7585_dual__order_Orefl,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),A) ).

% dual_order.refl
tff(fact_7586_dual__order_Orefl,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),A) ).

% dual_order.refl
tff(fact_7587_order__refl,axiom,
    ! [Xb: set_nat] : ord_less_eq_set_nat(Xb,Xb) ).

% order_refl
tff(fact_7588_order__refl,axiom,
    ! [Xb: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Xb) ).

% order_refl
tff(fact_7589_order__refl,axiom,
    ! [Xb: num] : aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Xb) ).

% order_refl
tff(fact_7590_order__refl,axiom,
    ! [Xb: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Xb) ).

% order_refl
tff(fact_7591_order__refl,axiom,
    ! [Xb: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xb) ).

% order_refl
tff(fact_7592__C5_Ohyps_C_I4_J,axiom,
    size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) ).

% "5.hyps"(4)
tff(fact_7593_DiffI,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B3: set_Extended_enat] :
      ( member_Extended_enat(C,A3)
     => ( ~ member_Extended_enat(C,B3)
       => member_Extended_enat(C,minus_925952699566721837d_enat(A3,B3)) ) ) ).

% DiffI
tff(fact_7594_DiffI,axiom,
    ! [C: complex,A3: set_complex,B3: set_complex] :
      ( member_complex(C,A3)
     => ( ~ member_complex(C,B3)
       => member_complex(C,minus_811609699411566653omplex(A3,B3)) ) ) ).

% DiffI
tff(fact_7595_DiffI,axiom,
    ! [C: real,A3: set_real,B3: set_real] :
      ( member_real(C,A3)
     => ( ~ member_real(C,B3)
       => member_real(C,minus_minus_set_real(A3,B3)) ) ) ).

% DiffI
tff(fact_7596_DiffI,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( member_int(C,A3)
     => ( ~ member_int(C,B3)
       => member_int(C,minus_minus_set_int(A3,B3)) ) ) ).

% DiffI
tff(fact_7597_DiffI,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( member_nat(C,A3)
     => ( ~ member_nat(C,B3)
       => member_nat(C,minus_minus_set_nat(A3,B3)) ) ) ).

% DiffI
tff(fact_7598_Diff__iff,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B3: set_Extended_enat] :
      ( member_Extended_enat(C,minus_925952699566721837d_enat(A3,B3))
    <=> ( member_Extended_enat(C,A3)
        & ~ member_Extended_enat(C,B3) ) ) ).

% Diff_iff
tff(fact_7599_Diff__iff,axiom,
    ! [C: complex,A3: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A3,B3))
    <=> ( member_complex(C,A3)
        & ~ member_complex(C,B3) ) ) ).

% Diff_iff
tff(fact_7600_Diff__iff,axiom,
    ! [C: real,A3: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A3,B3))
    <=> ( member_real(C,A3)
        & ~ member_real(C,B3) ) ) ).

% Diff_iff
tff(fact_7601_Diff__iff,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A3,B3))
    <=> ( member_int(C,A3)
        & ~ member_int(C,B3) ) ) ).

% Diff_iff
tff(fact_7602_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( member_nat(C,minus_minus_set_nat(A3,B3))
    <=> ( member_nat(C,A3)
        & ~ member_nat(C,B3) ) ) ).

% Diff_iff
tff(fact_7603_Diff__idemp,axiom,
    ! [A3: set_nat,B3: set_nat] : minus_minus_set_nat(minus_minus_set_nat(A3,B3),B3) = minus_minus_set_nat(A3,B3) ).

% Diff_idemp
tff(fact_7604_bit_Ocompl__eq__compl__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( aa_int_int(bit_ri7919022796975470100ot_int,Xb) = aa_int_int(bit_ri7919022796975470100ot_int,Y) )
    <=> ( Xb = Y ) ) ).

% bit.compl_eq_compl_iff
tff(fact_7605_bit_Odouble__compl,axiom,
    ! [Xb: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_ri7919022796975470100ot_int,Xb)) = Xb ).

% bit.double_compl
tff(fact_7606_bit_Oxor__compl__right,axiom,
    ! [Xb: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),aa_int_int(bit_ri7919022796975470100ot_int,Y)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(Xb),Y)) ).

% bit.xor_compl_right
tff(fact_7607_bit_Oxor__compl__left,axiom,
    ! [Xb: int,Y: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_ri7919022796975470100ot_int,Xb)),Y) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se6526347334894502574or_int(Xb),Y)) ).

% bit.xor_compl_left
tff(fact_7608_bit_Oconj__cancel__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(Xb),aa_int_int(bit_ri7919022796975470100ot_int,Xb)) = zero_zero_int ).

% bit.conj_cancel_right
tff(fact_7609_bit_Oconj__cancel__left,axiom,
    ! [Xb: int] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,Xb)),Xb) = zero_zero_int ).

% bit.conj_cancel_left
tff(fact_7610_bit_Ocompl__zero,axiom,
    bit_ri7632146776885996613nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_7611_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_7612_bit_Ocompl__one,axiom,
    bit_ri7632146776885996613nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_7613_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_7614_bit_Oxor__one__left,axiom,
    ! [Xb: code_integer] : bit_se3222712562003087583nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),Xb) = bit_ri7632146776885996613nteger(Xb) ).

% bit.xor_one_left
tff(fact_7615_bit_Oxor__one__left,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(uminus_uminus_int,one_one_int)),Xb) = aa_int_int(bit_ri7919022796975470100ot_int,Xb) ).

% bit.xor_one_left
tff(fact_7616_bit_Oxor__one__right,axiom,
    ! [Xb: code_integer] : bit_se3222712562003087583nteger(Xb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(Xb) ).

% bit.xor_one_right
tff(fact_7617_bit_Oxor__one__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,Xb) ).

% bit.xor_one_right
tff(fact_7618_bit_Oxor__cancel__left,axiom,
    ! [Xb: code_integer] : bit_se3222712562003087583nteger(bit_ri7632146776885996613nteger(Xb),Xb) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_left
tff(fact_7619_bit_Oxor__cancel__left,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(aa_int_int(bit_ri7919022796975470100ot_int,Xb)),Xb) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_left
tff(fact_7620_bit_Oxor__cancel__right,axiom,
    ! [Xb: code_integer] : bit_se3222712562003087583nteger(Xb,bit_ri7632146776885996613nteger(Xb)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_right
tff(fact_7621_bit_Oxor__cancel__right,axiom,
    ! [Xb: int] : aa_int_int(bit_se6526347334894502574or_int(Xb),aa_int_int(bit_ri7919022796975470100ot_int,Xb)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_right
tff(fact_7622_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_7623_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_7624_minus__not__numeral__eq,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Nb))) = numera6620942414471956472nteger(aa_num_num(inc,Nb)) ).

% minus_not_numeral_eq
tff(fact_7625_minus__not__numeral__eq,axiom,
    ! [Nb: num] : aa_int_int(uminus_uminus_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb)) ).

% minus_not_numeral_eq
tff(fact_7626_even__not__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),bit_ri7632146776885996613nteger(A))
    <=> ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(aa_num_num(bit0,one))),A) ) ).

% even_not_iff
tff(fact_7627_even__not__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(bit_ri7919022796975470100ot_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_not_iff
tff(fact_7628_push__bit__minus__one__eq__not__mask,axiom,
    ! [Nb: nat] : bit_se7788150548672797655nteger(Nb,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_ri7632146776885996613nteger(bit_se2119862282449309892nteger(Nb)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_7629_push__bit__minus__one__eq__not__mask,axiom,
    ! [Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Nb),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_7630_not__one__eq,axiom,
    bit_ri7632146776885996613nteger(one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_7631_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ).

% not_one_eq
tff(fact_7632_length__induct,axiom,
    ! [P: fun_list_VEBT_VEBT_o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys: list_VEBT_VEBT] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_s6755466524823107622T_VEBT(Ys)),size_s6755466524823107622T_VEBT(Xs2))
             => aa_list_VEBT_VEBT_o(P,Ys) )
         => aa_list_VEBT_VEBT_o(P,Xs2) )
     => aa_list_VEBT_VEBT_o(P,Xs) ) ).

% length_induct
tff(fact_7633_length__induct,axiom,
    ! [P: fun_list_o_o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys: list_o] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_o(Ys)),size_size_list_o(Xs2))
             => aa_list_o_o(P,Ys) )
         => aa_list_o_o(P,Xs2) )
     => aa_list_o_o(P,Xs) ) ).

% length_induct
tff(fact_7634_length__induct,axiom,
    ! [P: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys)),size_size_list_nat(Xs2))
             => aa_list_nat_o(P,Ys) )
         => aa_list_nat_o(P,Xs2) )
     => aa_list_nat_o(P,Xs) ) ).

% length_induct
tff(fact_7635_length__induct,axiom,
    ! [P: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys)),size_size_list_int(Xs2))
             => aa_list_int_o(P,Ys) )
         => aa_list_int_o(P,Xs2) )
     => aa_list_int_o(P,Xs) ) ).

% length_induct
tff(fact_7636_DiffE,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B3: set_Extended_enat] :
      ( member_Extended_enat(C,minus_925952699566721837d_enat(A3,B3))
     => ~ ( member_Extended_enat(C,A3)
         => member_Extended_enat(C,B3) ) ) ).

% DiffE
tff(fact_7637_DiffE,axiom,
    ! [C: complex,A3: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A3,B3))
     => ~ ( member_complex(C,A3)
         => member_complex(C,B3) ) ) ).

% DiffE
tff(fact_7638_DiffE,axiom,
    ! [C: real,A3: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A3,B3))
     => ~ ( member_real(C,A3)
         => member_real(C,B3) ) ) ).

% DiffE
tff(fact_7639_DiffE,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A3,B3))
     => ~ ( member_int(C,A3)
         => member_int(C,B3) ) ) ).

% DiffE
tff(fact_7640_DiffE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( member_nat(C,minus_minus_set_nat(A3,B3))
     => ~ ( member_nat(C,A3)
         => member_nat(C,B3) ) ) ).

% DiffE
tff(fact_7641_DiffD1,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B3: set_Extended_enat] :
      ( member_Extended_enat(C,minus_925952699566721837d_enat(A3,B3))
     => member_Extended_enat(C,A3) ) ).

% DiffD1
tff(fact_7642_DiffD1,axiom,
    ! [C: complex,A3: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A3,B3))
     => member_complex(C,A3) ) ).

% DiffD1
tff(fact_7643_DiffD1,axiom,
    ! [C: real,A3: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A3,B3))
     => member_real(C,A3) ) ).

% DiffD1
tff(fact_7644_DiffD1,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A3,B3))
     => member_int(C,A3) ) ).

% DiffD1
tff(fact_7645_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( member_nat(C,minus_minus_set_nat(A3,B3))
     => member_nat(C,A3) ) ).

% DiffD1
tff(fact_7646_DiffD2,axiom,
    ! [C: extended_enat,A3: set_Extended_enat,B3: set_Extended_enat] :
      ( member_Extended_enat(C,minus_925952699566721837d_enat(A3,B3))
     => ~ member_Extended_enat(C,B3) ) ).

% DiffD2
tff(fact_7647_DiffD2,axiom,
    ! [C: complex,A3: set_complex,B3: set_complex] :
      ( member_complex(C,minus_811609699411566653omplex(A3,B3))
     => ~ member_complex(C,B3) ) ).

% DiffD2
tff(fact_7648_DiffD2,axiom,
    ! [C: real,A3: set_real,B3: set_real] :
      ( member_real(C,minus_minus_set_real(A3,B3))
     => ~ member_real(C,B3) ) ).

% DiffD2
tff(fact_7649_DiffD2,axiom,
    ! [C: int,A3: set_int,B3: set_int] :
      ( member_int(C,minus_minus_set_int(A3,B3))
     => ~ member_int(C,B3) ) ).

% DiffD2
tff(fact_7650_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( member_nat(C,minus_minus_set_nat(A3,B3))
     => ~ member_nat(C,B3) ) ).

% DiffD2
tff(fact_7651_take__bit__not__iff,axiom,
    ! [Nb: nat,A: int,B2: int] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,B2)) )
    <=> ( aa_int_int(bit_se2923211474154528505it_int(Nb),A) = aa_int_int(bit_se2923211474154528505it_int(Nb),B2) ) ) ).

% take_bit_not_iff
tff(fact_7652_take__bit__not__take__bit,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se2923211474154528505it_int(Nb),A))) = aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) ).

% take_bit_not_take_bit
tff(fact_7653_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_7654_of__int__not__eq,axiom,
    ! [K: int] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,K)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(ring_1_of_int_int,K)) ).

% of_int_not_eq
tff(fact_7655_size__neq__size__imp__neq,axiom,
    ! [Xb: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xb) != size_s6755466524823107622T_VEBT(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_7656_size__neq__size__imp__neq,axiom,
    ! [Xb: list_o,Y: list_o] :
      ( ( size_size_list_o(Xb) != size_size_list_o(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_7657_size__neq__size__imp__neq,axiom,
    ! [Xb: list_nat,Y: list_nat] :
      ( ( size_size_list_nat(Xb) != size_size_list_nat(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_7658_size__neq__size__imp__neq,axiom,
    ! [Xb: list_int,Y: list_int] :
      ( ( size_size_list_int(Xb) != size_size_list_int(Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_7659_size__neq__size__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( ( aa_num_nat(size_size_num,Xb) != aa_num_nat(size_size_num,Y) )
     => ( Xb != Y ) ) ).

% size_neq_size_imp_neq
tff(fact_7660_of__int__not__numeral,axiom,
    ! [K: num] : aa_int_int(ring_1_of_int_int,aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,K)) ).

% of_int_not_numeral
tff(fact_7661_not__add__distrib,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B2) ).

% not_add_distrib
tff(fact_7662_not__diff__distrib,axiom,
    ! [A: int,B2: int] : aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),B2) ).

% not_diff_distrib
tff(fact_7663_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = plus_p5714425477246183910nteger(bit_ri7632146776885996613nteger(A),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_7664_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_7665_not__eq__complement,axiom,
    ! [A: code_integer] : bit_ri7632146776885996613nteger(A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),one_one_Code_integer) ).

% not_eq_complement
tff(fact_7666_not__eq__complement,axiom,
    ! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).

% not_eq_complement
tff(fact_7667_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = bit_ri7632146776885996613nteger(minus_8373710615458151222nteger(A,one_one_Code_integer)) ).

% minus_eq_not_minus_1
tff(fact_7668_minus__eq__not__minus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).

% minus_eq_not_minus_1
tff(fact_7669_not__int__def,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,K)),one_one_int) ).

% not_int_def
tff(fact_7670_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_7671_disjunctive__diff,axiom,
    ! [B2: int,A: int] :
      ( ! [N2: nat] :
          ( aa_nat_o(bit_se1146084159140164899it_int(B2),N2)
         => aa_nat_o(bit_se1146084159140164899it_int(A),N2) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,B2)) ) ) ).

% disjunctive_diff
tff(fact_7672_nle__le,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_7673_nle__le,axiom,
    ! [A: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_7674_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_7675_nle__le,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ( B2 != A ) ) ) ).

% nle_le
tff(fact_7676_le__cases3,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
       => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb) )
     => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
         => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Zb) )
       => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Zb)
           => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Zb),Y) )
         => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Zb),Y)
             => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) )
           => ( ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb)
               => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Zb),Xb) )
             => ~ ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Zb),Xb)
                 => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_7677_le__cases3,axiom,
    ! [Xb: num,Y: num,Zb: num] :
      ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
       => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Zb) )
     => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
         => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Zb) )
       => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Zb)
           => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Zb),Y) )
         => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Zb),Y)
             => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) )
           => ( ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Zb)
               => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Zb),Xb) )
             => ~ ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Zb),Xb)
                 => ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_7678_le__cases3,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
       => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Zb) )
     => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Zb) )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Zb)
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Zb),Y) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Zb),Y)
             => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) )
           => ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Zb)
               => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Zb),Xb) )
             => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Zb),Xb)
                 => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_7679_le__cases3,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
       => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Zb) )
     => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Zb) )
       => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Zb)
           => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),Y) )
         => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),Y)
             => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) )
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Zb)
               => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),Xb) )
             => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Zb),Xb)
                 => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ) ) ) ) ) ).

% le_cases3
tff(fact_7680_order__class_Oorder__eq__iff,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( Xb = Y )
    <=> ( ord_less_eq_set_nat(Xb,Y)
        & ord_less_eq_set_nat(Y,Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_7681_order__class_Oorder__eq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_7682_order__class_Oorder__eq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_7683_order__class_Oorder__eq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_7684_order__class_Oorder__eq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% order_class.order_eq_iff
tff(fact_7685_ord__eq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ( A = B2 )
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_eq_set_nat(A,C) ) ) ).

% ord_eq_le_trans
tff(fact_7686_ord__eq__le__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_7687_ord__eq__le__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_7688_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_7689_ord__eq__le__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_eq_le_trans
tff(fact_7690_ord__le__eq__trans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
     => ( ( B2 = C )
       => ord_less_eq_set_nat(A,C) ) ) ).

% ord_le_eq_trans
tff(fact_7691_ord__le__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( B2 = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_7692_ord__le__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( B2 = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_7693_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( B2 = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_7694_ord__le__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( ( B2 = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% ord_le_eq_trans
tff(fact_7695_order__antisym,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ( ord_less_eq_set_nat(Y,Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_7696_order__antisym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_7697_order__antisym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_7698_order__antisym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_7699_order__antisym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
       => ( Xb = Y ) ) ) ).

% order_antisym
tff(fact_7700_order_Otrans,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_eq_set_nat(A,C) ) ) ).

% order.trans
tff(fact_7701_order_Otrans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),C) ) ) ).

% order.trans
tff(fact_7702_order_Otrans,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),C) ) ) ).

% order.trans
tff(fact_7703_order_Otrans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% order.trans
tff(fact_7704_order_Otrans,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% order.trans
tff(fact_7705_order__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Zb: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ( ord_less_eq_set_nat(Y,Zb)
       => ord_less_eq_set_nat(Xb,Zb) ) ) ).

% order_trans
tff(fact_7706_order__trans,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Zb) ) ) ).

% order_trans
tff(fact_7707_order__trans,axiom,
    ! [Xb: num,Y: num,Zb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Zb)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Zb) ) ) ).

% order_trans
tff(fact_7708_order__trans,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Zb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Zb) ) ) ).

% order_trans
tff(fact_7709_order__trans,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Zb) ) ) ).

% order_trans
tff(fact_7710_linorder__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B2: rat] :
      ( ! [A4: rat,B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A4),B4)
         => aa_rat_o(aa_rat_fun_rat_o(P,A4),B4) )
     => ( ! [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),B2) ) ) ).

% linorder_wlog
tff(fact_7711_linorder__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B2: num] :
      ( ! [A4: num,B4: num] :
          ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A4),B4)
         => aa_num_o(aa_num_fun_num_o(P,A4),B4) )
     => ( ! [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),B2) ) ) ).

% linorder_wlog
tff(fact_7712_linorder__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A4),B4)
         => aa_nat_o(aa_nat_fun_nat_o(P,A4),B4) )
     => ( ! [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),B2) ) ) ).

% linorder_wlog
tff(fact_7713_linorder__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B2: int] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A4),B4)
         => aa_int_o(aa_int_fun_int_o(P,A4),B4) )
     => ( ! [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),B2) ) ) ).

% linorder_wlog
tff(fact_7714_dual__order_Oeq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
    <=> ( ord_less_eq_set_nat(B2,A)
        & ord_less_eq_set_nat(A,B2) ) ) ).

% dual_order.eq_iff
tff(fact_7715_dual__order_Oeq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_7716_dual__order_Oeq__iff,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_7717_dual__order_Oeq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_7718_dual__order_Oeq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% dual_order.eq_iff
tff(fact_7719_dual__order_Oantisym,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ord_less_eq_set_nat(B2,A)
     => ( ord_less_eq_set_nat(A,B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_7720_dual__order_Oantisym,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_7721_dual__order_Oantisym,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_7722_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_7723_dual__order_Oantisym,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
       => ( A = B2 ) ) ) ).

% dual_order.antisym
tff(fact_7724_dual__order_Otrans,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(B2,A)
     => ( ord_less_eq_set_nat(C,B2)
       => ord_less_eq_set_nat(C,A) ) ) ).

% dual_order.trans
tff(fact_7725_dual__order_Otrans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),A) ) ) ).

% dual_order.trans
tff(fact_7726_dual__order_Otrans,axiom,
    ! [B2: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),A) ) ) ).

% dual_order.trans
tff(fact_7727_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% dual_order.trans
tff(fact_7728_dual__order_Otrans,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% dual_order.trans
tff(fact_7729_antisym,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
     => ( ord_less_eq_set_nat(B2,A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_7730_antisym,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_7731_antisym,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_7732_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_7733_antisym,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
       => ( A = B2 ) ) ) ).

% antisym
tff(fact_7734_Orderings_Oorder__eq__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A = B2 )
    <=> ( ord_less_eq_set_nat(A,B2)
        & ord_less_eq_set_nat(B2,A) ) ) ).

% Orderings.order_eq_iff
tff(fact_7735_Orderings_Oorder__eq__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( A = B2 )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_7736_Orderings_Oorder__eq__iff,axiom,
    ! [A: num,B2: num] :
      ( ( A = B2 )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_7737_Orderings_Oorder__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( A = B2 )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_7738_Orderings_Oorder__eq__iff,axiom,
    ! [A: int,B2: int] :
      ( ( A = B2 )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% Orderings.order_eq_iff
tff(fact_7739_order__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7740_order__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7741_order__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7742_order__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7743_order__subst1,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_rat_num(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_7744_order__subst1,axiom,
    ! [A: num,F: fun_num_num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_num_num(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_7745_order__subst1,axiom,
    ! [A: num,F: fun_nat_num,B2: nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_7746_order__subst1,axiom,
    ! [A: num,F: fun_int_num,B2: int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_int_num(F,X3)),aa_int_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_int_num(F,C)) ) ) ) ).

% order_subst1
tff(fact_7747_order__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_rat_nat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7748_order__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_subst1
tff(fact_7749_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B2)),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_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7750_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_rat_num(F,B2)),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_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7751_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_rat_nat(F,B2)),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_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7752_order__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_rat_int(F,B2)),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_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7753_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B2)),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_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7754_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_num_num(F,B2)),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_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7755_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,B2)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7756_order__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,B2)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7757_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B2)),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7758_order__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,B2)),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% order_subst2
tff(fact_7759_order__eq__refl,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( Xb = Y )
     => ord_less_eq_set_nat(Xb,Y) ) ).

% order_eq_refl
tff(fact_7760_order__eq__refl,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb = Y )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_eq_refl
tff(fact_7761_order__eq__refl,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb = Y )
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_eq_refl
tff(fact_7762_order__eq__refl,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb = Y )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_eq_refl
tff(fact_7763_order__eq__refl,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb = Y )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_eq_refl
tff(fact_7764_linorder__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_linear
tff(fact_7765_linorder__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_linear
tff(fact_7766_linorder__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_linear
tff(fact_7767_linorder__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_linear
tff(fact_7768_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_rat,A),aa_rat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7769_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C: rat] :
      ( ( A = aa_rat_num(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_num,A),aa_rat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7770_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_nat,A),aa_rat_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7771_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C: rat] :
      ( ( A = aa_rat_int(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_eq_int,A),aa_rat_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7772_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C: num] :
      ( ( A = aa_num_rat(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_eq_rat,A),aa_num_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7773_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_num_num,B2: num,C: num] :
      ( ( A = aa_num_num(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_eq_num,A),aa_num_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7774_ord__eq__le__subst,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C: num] :
      ( ( A = aa_num_nat(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_num_nat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7775_ord__eq__le__subst,axiom,
    ! [A: int,F: fun_num_int,B2: num,C: num] :
      ( ( A = aa_num_int(F,B2) )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_num_int(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7776_ord__eq__le__subst,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C: nat] :
      ( ( A = aa_nat_rat(F,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7777_ord__eq__le__subst,axiom,
    ! [A: num,F: fun_nat_num,B2: nat,C: nat] :
      ( ( A = aa_nat_num(F,B2) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),aa_nat_num(F,C)) ) ) ) ).

% ord_eq_le_subst
tff(fact_7778_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_rat(F,B2) = 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_eq_rat,aa_rat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7779_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_num(F,B2) = 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_eq_num,aa_rat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7780_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_nat(F,B2) = 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_eq_nat,aa_rat_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7781_ord__le__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( aa_rat_int(F,B2) = 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_eq_int,aa_rat_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7782_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_rat(F,B2) = 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_eq_rat,aa_num_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7783_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_num(F,B2) = 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_eq_num,aa_num_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7784_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_nat(F,B2) = C )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7785_ord__le__eq__subst,axiom,
    ! [A: num,B2: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( aa_num_int(F,B2) = C )
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7786_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( aa_nat_rat(F,B2) = C )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,X3)),aa_nat_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7787_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: fun_nat_num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( aa_nat_num(F,B2) = C )
       => ( ! [X3: nat,Y3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X3),Y3)
             => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,X3)),aa_nat_num(F,Y3)) )
         => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,aa_nat_num(F,A)),C) ) ) ) ).

% ord_le_eq_subst
tff(fact_7788_linorder__le__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_7789_linorder__le__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_le_cases
tff(fact_7790_linorder__le__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_le_cases
tff(fact_7791_linorder__le__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_le_cases
tff(fact_7792_order__antisym__conv,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( ord_less_eq_set_nat(Y,Xb)
     => ( ord_less_eq_set_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_7793_order__antisym__conv,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_7794_order__antisym__conv,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_7795_order__antisym__conv,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_7796_order__antisym__conv,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% order_antisym_conv
tff(fact_7797_lt__ex,axiom,
    ! [Xb: real] :
    ? [Y3: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Y3),Xb) ).

% lt_ex
tff(fact_7798_lt__ex,axiom,
    ! [Xb: rat] :
    ? [Y3: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y3),Xb) ).

% lt_ex
tff(fact_7799_lt__ex,axiom,
    ! [Xb: int] :
    ? [Y3: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Y3),Xb) ).

% lt_ex
tff(fact_7800_gt__ex,axiom,
    ! [Xb: real] :
    ? [X_1: real] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X_1) ).

% gt_ex
tff(fact_7801_gt__ex,axiom,
    ! [Xb: rat] :
    ? [X_1: rat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),X_1) ).

% gt_ex
tff(fact_7802_gt__ex,axiom,
    ! [Xb: nat] :
    ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X_1) ).

% gt_ex
tff(fact_7803_gt__ex,axiom,
    ! [Xb: int] :
    ? [X_1: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),X_1) ).

% gt_ex
tff(fact_7804_dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Z: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Z)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),Y) ) ) ).

% dense
tff(fact_7805_dense,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ? [Z: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Z)
          & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Z),Y) ) ) ).

% dense
tff(fact_7806_less__imp__neq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_7807_less__imp__neq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_7808_less__imp__neq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_7809_less__imp__neq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_7810_less__imp__neq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% less_imp_neq
tff(fact_7811_order_Oasym,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% order.asym
tff(fact_7812_order_Oasym,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% order.asym
tff(fact_7813_order_Oasym,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A) ) ).

% order.asym
tff(fact_7814_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A) ) ).

% order.asym
tff(fact_7815_order_Oasym,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% order.asym
tff(fact_7816_ord__eq__less__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( ( A = B2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_7817_ord__eq__less__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( ( A = B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_7818_ord__eq__less__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( ( A = B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_7819_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( ( A = B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_7820_ord__eq__less__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( ( A = B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_eq_less_trans
tff(fact_7821_ord__less__eq__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( B2 = C )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_7822_ord__less__eq__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( B2 = C )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_7823_ord__less__eq__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( ( B2 = C )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_7824_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( ( B2 = C )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_7825_ord__less__eq__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( ( B2 = C )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% ord_less_eq_trans
tff(fact_7826_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_7827_antisym__conv3,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_7828_antisym__conv3,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_7829_antisym__conv3,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_7830_antisym__conv3,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_7831_antisym__conv3,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv3
tff(fact_7832_linorder__cases,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ( Xb != Y )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_cases
tff(fact_7833_linorder__cases,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ( Xb != Y )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_7834_linorder__cases,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( ( Xb != Y )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_cases
tff(fact_7835_linorder__cases,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( ( Xb != Y )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_cases
tff(fact_7836_linorder__cases,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( ( Xb != Y )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_cases
tff(fact_7837_dual__order_Oasym,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ).

% dual_order.asym
tff(fact_7838_dual__order_Oasym,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ).

% dual_order.asym
tff(fact_7839_dual__order_Oasym,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ).

% dual_order.asym
tff(fact_7840_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ).

% dual_order.asym
tff(fact_7841_dual__order_Oasym,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ).

% dual_order.asym
tff(fact_7842_dual__order_Oirrefl,axiom,
    ! [A: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),A) ).

% dual_order.irrefl
tff(fact_7843_dual__order_Oirrefl,axiom,
    ! [A: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),A) ).

% dual_order.irrefl
tff(fact_7844_dual__order_Oirrefl,axiom,
    ! [A: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),A) ).

% dual_order.irrefl
tff(fact_7845_dual__order_Oirrefl,axiom,
    ! [A: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),A) ).

% dual_order.irrefl
tff(fact_7846_dual__order_Oirrefl,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),A) ).

% dual_order.irrefl
tff(fact_7847_exists__least__iff,axiom,
    ! [P: fun_nat_o] :
      ( ? [X_13: nat] : aa_nat_o(P,X_13)
    <=> ? [N4: nat] :
          ( aa_nat_o(P,N4)
          & ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N4)
             => ~ aa_nat_o(P,M6) ) ) ) ).

% exists_least_iff
tff(fact_7848_linorder__less__wlog,axiom,
    ! [P: fun_real_fun_real_o,A: real,B2: 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),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_7849_linorder__less__wlog,axiom,
    ! [P: fun_rat_fun_rat_o,A: rat,B2: 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),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_7850_linorder__less__wlog,axiom,
    ! [P: fun_num_fun_num_o,A: num,B2: 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),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_7851_linorder__less__wlog,axiom,
    ! [P: fun_nat_fun_nat_o,A: nat,B2: 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),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_7852_linorder__less__wlog,axiom,
    ! [P: fun_int_fun_int_o,A: int,B2: 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),B2) ) ) ) ).

% linorder_less_wlog
tff(fact_7853_order_Ostrict__trans,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans
tff(fact_7854_order_Ostrict__trans,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans
tff(fact_7855_order_Ostrict__trans,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans
tff(fact_7856_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans
tff(fact_7857_order_Ostrict__trans,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans
tff(fact_7858_not__less__iff__gr__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7859_not__less__iff__gr__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7860_not__less__iff__gr__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7861_not__less__iff__gr__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7862_not__less__iff__gr__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
        | ( Xb = Y ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_7863_dual__order_Ostrict__trans,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_7864_dual__order_Ostrict__trans,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_7865_dual__order_Ostrict__trans,axiom,
    ! [B2: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_7866_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_7867_dual__order_Ostrict__trans,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans
tff(fact_7868_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_7869_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_7870_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_7871_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_7872_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
tff(fact_7873_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_7874_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_7875_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_7876_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_7877_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_7878_linorder__neqE,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_7879_linorder__neqE,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_7880_linorder__neqE,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_7881_linorder__neqE,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_7882_linorder__neqE,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neqE
tff(fact_7883_order__less__asym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_asym
tff(fact_7884_order__less__asym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_asym
tff(fact_7885_order__less__asym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_asym
tff(fact_7886_order__less__asym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_asym
tff(fact_7887_order__less__asym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_asym
tff(fact_7888_linorder__neq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != Y )
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_7889_linorder__neq__iff,axiom,
    ! [Xb: rat,Y: rat] :
      ( ( Xb != Y )
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_7890_linorder__neq__iff,axiom,
    ! [Xb: num,Y: num] :
      ( ( Xb != Y )
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_7891_linorder__neq__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_7892_linorder__neq__iff,axiom,
    ! [Xb: int,Y: int] :
      ( ( Xb != Y )
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ) ).

% linorder_neq_iff
tff(fact_7893_order__less__asym_H,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A) ) ).

% order_less_asym'
tff(fact_7894_order__less__asym_H,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A) ) ).

% order_less_asym'
tff(fact_7895_order__less__asym_H,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A) ) ).

% order_less_asym'
tff(fact_7896_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A) ) ).

% order_less_asym'
tff(fact_7897_order__less__asym_H,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ).

% order_less_asym'
tff(fact_7898_order__less__trans,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Zb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Zb) ) ) ).

% order_less_trans
tff(fact_7899_order__less__trans,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Zb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Zb) ) ) ).

% order_less_trans
tff(fact_7900_order__less__trans,axiom,
    ! [Xb: num,Y: num,Zb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Zb)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Zb) ) ) ).

% order_less_trans
tff(fact_7901_order__less__trans,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Zb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Zb) ) ) ).

% order_less_trans
tff(fact_7902_order__less__trans,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Zb) ) ) ).

% order_less_trans
tff(fact_7903_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( ( A = aa_real_real(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7904_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C: real] :
      ( ( A = aa_real_rat(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7905_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_real_num,B2: real,C: real] :
      ( ( A = aa_real_num(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7906_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_real_nat,B2: real,C: real] :
      ( ( A = aa_real_nat(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7907_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_real_int,B2: real,C: real] :
      ( ( A = aa_real_int(F,B2) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7908_ord__eq__less__subst,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C: rat] :
      ( ( A = aa_rat_real(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7909_ord__eq__less__subst,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( ( A = aa_rat_rat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7910_ord__eq__less__subst,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C: rat] :
      ( ( A = aa_rat_num(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7911_ord__eq__less__subst,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C: rat] :
      ( ( A = aa_rat_nat(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7912_ord__eq__less__subst,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C: rat] :
      ( ( A = aa_rat_int(F,B2) )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_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)) ) ) ) ).

% ord_eq_less_subst
tff(fact_7913_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,B2) = 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_7914_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_rat(F,B2) = 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_7915_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_num(F,B2) = 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_7916_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_nat(F,B2) = 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_7917_ord__less__eq__subst,axiom,
    ! [A: real,B2: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_int(F,B2) = 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_7918_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_real(F,B2) = 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_7919_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_rat(F,B2) = 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_7920_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_num(F,B2) = 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_7921_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_nat(F,B2) = 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_7922_ord__less__eq__subst,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( ( aa_rat_int(F,B2) = C )
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_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) ) ) ) ).

% ord_less_eq_subst
tff(fact_7923_order__less__irrefl,axiom,
    ! [Xb: real] : ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Xb) ).

% order_less_irrefl
tff(fact_7924_order__less__irrefl,axiom,
    ! [Xb: rat] : ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Xb) ).

% order_less_irrefl
tff(fact_7925_order__less__irrefl,axiom,
    ! [Xb: num] : ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Xb) ).

% order_less_irrefl
tff(fact_7926_order__less__irrefl,axiom,
    ! [Xb: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Xb) ).

% order_less_irrefl
tff(fact_7927_order__less__irrefl,axiom,
    ! [Xb: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xb) ).

% order_less_irrefl
tff(fact_7928_order__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_real_real(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7929_order__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7930_order__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),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_7931_order__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_nat_real(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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_7932_order__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_int_real(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),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_7933_order__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_real_rat(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_7934_order__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_7935_order__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),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_7936_order__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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_7937_order__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_less_subst1
tff(fact_7938_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B2)),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_7939_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_real_rat(F,B2)),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_7940_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_real_num(F,B2)),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_7941_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_real_nat(F,B2)),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_7942_order__less__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_real_int(F,B2)),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_7943_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B2)),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_7944_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B2)),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_7945_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B2)),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_7946_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B2)),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_7947_order__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B2)),C)
       => ( ! [X3: rat,Y3: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_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_less_subst2
tff(fact_7948_order__less__not__sym,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_not_sym
tff(fact_7949_order__less__not__sym,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_7950_order__less__not__sym,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_not_sym
tff(fact_7951_order__less__not__sym,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_not_sym
tff(fact_7952_order__less__not__sym,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_not_sym
tff(fact_7953_order__less__imp__triv,axiom,
    ! [Xb: real,Y: real,P: $o] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_7954_order__less__imp__triv,axiom,
    ! [Xb: rat,Y: rat,P: $o] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_7955_order__less__imp__triv,axiom,
    ! [Xb: num,Y: num,P: $o] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_7956_order__less__imp__triv,axiom,
    ! [Xb: nat,Y: nat,P: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_7957_order__less__imp__triv,axiom,
    ! [Xb: int,Y: int,P: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb)
       => (P) ) ) ).

% order_less_imp_triv
tff(fact_7958_linorder__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      | ( Xb = Y )
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_less_linear
tff(fact_7959_linorder__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      | ( Xb = Y )
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_7960_linorder__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      | ( Xb = Y )
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_less_linear
tff(fact_7961_linorder__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      | ( Xb = Y )
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_less_linear
tff(fact_7962_linorder__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      | ( Xb = Y )
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_less_linear
tff(fact_7963_order__less__imp__not__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_7964_order__less__imp__not__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_7965_order__less__imp__not__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_7966_order__less__imp__not__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_7967_order__less__imp__not__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Xb != Y ) ) ).

% order_less_imp_not_eq
tff(fact_7968_order__less__imp__not__eq2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_7969_order__less__imp__not__eq2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_7970_order__less__imp__not__eq2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_7971_order__less__imp__not__eq2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_7972_order__less__imp__not__eq2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( Y != Xb ) ) ).

% order_less_imp_not_eq2
tff(fact_7973_order__less__imp__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_7974_order__less__imp__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_7975_order__less__imp__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_7976_order__less__imp__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_7977_order__less__imp__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% order_less_imp_not_less
tff(fact_7978_take__bit__not__eq__mask__diff,axiom,
    ! [Nb: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(Nb),aa_int_int(bit_ri7919022796975470100ot_int,A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)),aa_int_int(bit_se2923211474154528505it_int(Nb),A)) ).

% take_bit_not_eq_mask_diff
tff(fact_7979_minus__numeral__inc__eq,axiom,
    ! [Nb: num] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(inc,Nb))) = bit_ri7632146776885996613nteger(numera6620942414471956472nteger(Nb)) ).

% minus_numeral_inc_eq
tff(fact_7980_minus__numeral__inc__eq,axiom,
    ! [Nb: num] : aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(inc,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)) ).

% minus_numeral_inc_eq
tff(fact_7981_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_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_7982_not__int__div__2,axiom,
    ! [K: int] : aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))) ).

% not_int_div_2
tff(fact_7983_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_7984_not__numeral__Bit0__eq,axiom,
    ! [Nb: num] : bit_ri7632146776885996613nteger(numera6620942414471956472nteger(aa_num_num(bit0,Nb))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(aa_num_num(bit1,Nb))) ).

% not_numeral_Bit0_eq
tff(fact_7985_not__numeral__Bit0__eq,axiom,
    ! [Nb: num] : aa_int_int(bit_ri7919022796975470100ot_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(bit1,Nb))) ).

% not_numeral_Bit0_eq
tff(fact_7986_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_7987_and__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(4)
tff(fact_7988_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(aa_int_fun_int_int(minus_minus_int,K),one_one_int))),Nb) ) ).

% bit_minus_int_iff
tff(fact_7989_take__bit__not__mask__eq__0,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(bit_se2923211474154528505it_int(Mb),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb))) = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
tff(fact_7990_push__bit__mask__eq,axiom,
    ! [Mb: nat,Nb: nat] : aa_int_int(bit_se545348938243370406it_int(Mb),aa_nat_int(bit_se2000444600071755411sk_int,Nb)) = aa_int_int(bit_se725231765392027082nd_int(aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Mb))) ).

% push_bit_mask_eq
tff(fact_7991_unset__bit__eq__and__not,axiom,
    ! [Nb: nat,A: int] : aa_int_int(aa_nat_fun_int_int(bit_se4203085406695923979it_int,Nb),A) = aa_int_int(bit_se725231765392027082nd_int(A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(Nb),one_one_int))) ).

% unset_bit_eq_and_not
tff(fact_7992_and__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(5)
tff(fact_7993_and__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb)) ).

% and_not_numerals(7)
tff(fact_7994_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_7995_and__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(9)
tff(fact_7996_and__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% and_not_numerals(6)
tff(fact_7997_bit__not__iff__eq,axiom,
    ! [A: int,Nb: nat] :
      ( aa_nat_o(bit_se1146084159140164899it_int(aa_int_int(bit_ri7919022796975470100ot_int,A)),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(bit_se1146084159140164899it_int(A),Nb) ) ) ).

% bit_not_iff_eq
tff(fact_7998_minus__exp__eq__not__mask,axiom,
    ! [Nb: nat] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(aa_num_num(bit0,one)),Nb)) = bit_ri7632146776885996613nteger(bit_se2119862282449309892nteger(Nb)) ).

% minus_exp_eq_not_mask
tff(fact_7999_minus__exp__eq__not__mask,axiom,
    ! [Nb: nat] : 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_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,Nb)) ).

% minus_exp_eq_not_mask
tff(fact_8000_CauchyD,axiom,
    ! [X5: fun_nat_complex,E: real] :
      ( topolo6517432010174082258omplex(X5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ? [M8: nat] :
          ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M4)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N5)
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M4)),aa_nat_complex(X5,N5)))),E) ) ) ) ) ).

% CauchyD
tff(fact_8001_CauchyD,axiom,
    ! [X5: fun_nat_real,E: real] :
      ( topolo4055970368930404560y_real(X5)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E)
       => ? [M8: nat] :
          ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),M4)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M8),N5)
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M4)),aa_nat_real(X5,N5)))),E) ) ) ) ) ).

% CauchyD
tff(fact_8002_CauchyI,axiom,
    ! [X5: fun_nat_complex] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => ? [M9: nat] :
            ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M3)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M3)),aa_nat_complex(X5,N2)))),E2) ) ) )
     => topolo6517432010174082258omplex(X5) ) ).

% CauchyI
tff(fact_8003_CauchyI,axiom,
    ! [X5: fun_nat_real] :
      ( ! [E2: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
         => ? [M9: nat] :
            ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),M3)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M9),N2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M3)),aa_nat_real(X5,N2)))),E2) ) ) )
     => topolo4055970368930404560y_real(X5) ) ).

% CauchyI
tff(fact_8004_Cauchy__iff,axiom,
    ! [X5: fun_nat_complex] :
      ( topolo6517432010174082258omplex(X5)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M7: nat] :
            ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M6)
             => ! [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_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(aa_nat_complex(X5,M6)),aa_nat_complex(X5,N4)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_8005_Cauchy__iff,axiom,
    ! [X5: fun_nat_real] :
      ( topolo4055970368930404560y_real(X5)
    <=> ! [E3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E3)
         => ? [M7: nat] :
            ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M7),M6)
             => ! [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(real_V7735802525324610683m_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(X5,M6)),aa_nat_real(X5,N4)))),E3) ) ) ) ) ).

% Cauchy_iff
tff(fact_8006_leD,axiom,
    ! [Y: real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% leD
tff(fact_8007_leD,axiom,
    ! [Y: set_nat,Xb: set_nat] :
      ( ord_less_eq_set_nat(Y,Xb)
     => ~ ord_less_set_nat(Xb,Y) ) ).

% leD
tff(fact_8008_leD,axiom,
    ! [Y: rat,Xb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% leD
tff(fact_8009_leD,axiom,
    ! [Y: num,Xb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% leD
tff(fact_8010_leD,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% leD
tff(fact_8011_leD,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% leD
tff(fact_8012_leI,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% leI
tff(fact_8013_leI,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% leI
tff(fact_8014_leI,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% leI
tff(fact_8015_leI,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% leI
tff(fact_8016_leI,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% leI
tff(fact_8017_nless__le,axiom,
    ! [A: real,B2: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8018_nless__le,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ~ ord_less_set_nat(A,B2)
    <=> ( ~ ord_less_eq_set_nat(A,B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8019_nless__le,axiom,
    ! [A: rat,B2: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8020_nless__le,axiom,
    ! [A: num,B2: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8021_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8022_nless__le,axiom,
    ! [A: int,B2: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        | ( A = B2 ) ) ) ).

% nless_le
tff(fact_8023_antisym__conv1,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8024_antisym__conv1,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ~ ord_less_set_nat(Xb,Y)
     => ( ord_less_eq_set_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8025_antisym__conv1,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8026_antisym__conv1,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8027_antisym__conv1,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8028_antisym__conv1,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv1
tff(fact_8029_antisym__conv2,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8030_antisym__conv2,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ( ~ ord_less_set_nat(Xb,Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8031_antisym__conv2,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8032_antisym__conv2,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8033_antisym__conv2,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8034_antisym__conv2,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
      <=> ( Xb = Y ) ) ) ).

% antisym_conv2
tff(fact_8035_dense__ge,axiom,
    ! [Zb: real,Y: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Zb),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),Zb) ) ).

% dense_ge
tff(fact_8036_dense__ge,axiom,
    ! [Zb: rat,Y: rat] :
      ( ! [X3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Zb),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),Zb) ) ).

% dense_ge
tff(fact_8037_dense__le,axiom,
    ! [Y: real,Zb: 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),Zb) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Zb) ) ).

% dense_le
tff(fact_8038_dense__le,axiom,
    ! [Y: rat,Zb: 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),Zb) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb) ) ).

% dense_le
tff(fact_8039_less__le__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_8040_less__le__not__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_set_nat(Xb,Y)
    <=> ( ord_less_eq_set_nat(Xb,Y)
        & ~ ord_less_eq_set_nat(Y,Xb) ) ) ).

% less_le_not_le
tff(fact_8041_less__le__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_8042_less__le__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_8043_less__le__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_8044_less__le__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ) ).

% less_le_not_le
tff(fact_8045_not__le__imp__less,axiom,
    ! [Y: real,Xb: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ).

% not_le_imp_less
tff(fact_8046_not__le__imp__less,axiom,
    ! [Y: rat,Xb: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_8047_not__le__imp__less,axiom,
    ! [Y: num,Xb: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb)
     => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y) ) ).

% not_le_imp_less
tff(fact_8048_not__le__imp__less,axiom,
    ! [Y: nat,Xb: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% not_le_imp_less
tff(fact_8049_not__le__imp__less,axiom,
    ! [Y: int,Xb: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y) ) ).

% not_le_imp_less
tff(fact_8050_order_Oorder__iff__strict,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8051_order_Oorder__iff__strict,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
    <=> ( ord_less_set_nat(A,B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8052_order_Oorder__iff__strict,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8053_order_Oorder__iff__strict,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8054_order_Oorder__iff__strict,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8055_order_Oorder__iff__strict,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
        | ( A = B2 ) ) ) ).

% order.order_iff_strict
tff(fact_8056_order_Ostrict__iff__order,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8057_order_Ostrict__iff__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A,B2)
    <=> ( ord_less_eq_set_nat(A,B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8058_order_Ostrict__iff__order,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8059_order_Ostrict__iff__order,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8060_order_Ostrict__iff__order,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8061_order_Ostrict__iff__order,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & ( A != B2 ) ) ) ).

% order.strict_iff_order
tff(fact_8062_order_Ostrict__trans1,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans1
tff(fact_8063_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
     => ( ord_less_set_nat(B2,C)
       => ord_less_set_nat(A,C) ) ) ).

% order.strict_trans1
tff(fact_8064_order_Ostrict__trans1,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans1
tff(fact_8065_order_Ostrict__trans1,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans1
tff(fact_8066_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans1
tff(fact_8067_order_Ostrict__trans1,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans1
tff(fact_8068_order_Ostrict__trans2,axiom,
    ! [A: real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),C)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),C) ) ) ).

% order.strict_trans2
tff(fact_8069_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat] :
      ( ord_less_set_nat(A,B2)
     => ( ord_less_eq_set_nat(B2,C)
       => ord_less_set_nat(A,C) ) ) ).

% order.strict_trans2
tff(fact_8070_order_Ostrict__trans2,axiom,
    ! [A: rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),C)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),C) ) ) ).

% order.strict_trans2
tff(fact_8071_order_Ostrict__trans2,axiom,
    ! [A: num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),C) ) ) ).

% order.strict_trans2
tff(fact_8072_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% order.strict_trans2
tff(fact_8073_order_Ostrict__trans2,axiom,
    ! [A: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% order.strict_trans2
tff(fact_8074_order_Ostrict__iff__not,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_8075_order_Ostrict__iff__not,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A,B2)
    <=> ( ord_less_eq_set_nat(A,B2)
        & ~ ord_less_eq_set_nat(B2,A) ) ) ).

% order.strict_iff_not
tff(fact_8076_order_Ostrict__iff__not,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_8077_order_Ostrict__iff__not,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_8078_order_Ostrict__iff__not,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_8079_order_Ostrict__iff__not,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% order.strict_iff_not
tff(fact_8080_dense__ge__bounded,axiom,
    ! [Zb: real,Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Zb),Xb)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Zb),W2)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,W2),Xb)
             => 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),Zb) ) ) ).

% dense_ge_bounded
tff(fact_8081_dense__ge__bounded,axiom,
    ! [Zb: rat,Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Zb),Xb)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Zb),W2)
           => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,W2),Xb)
             => 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),Zb) ) ) ).

% dense_ge_bounded
tff(fact_8082_dense__le__bounded,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( ! [W2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),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),Zb) ) )
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Zb) ) ) ).

% dense_le_bounded
tff(fact_8083_dense__le__bounded,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( ! [W2: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),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),Zb) ) )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb) ) ) ).

% dense_le_bounded
tff(fact_8084_dual__order_Oorder__iff__strict,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8085_dual__order_Oorder__iff__strict,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ord_less_eq_set_nat(B2,A)
    <=> ( ord_less_set_nat(B2,A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8086_dual__order_Oorder__iff__strict,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8087_dual__order_Oorder__iff__strict,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8088_dual__order_Oorder__iff__strict,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8089_dual__order_Oorder__iff__strict,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
        | ( A = B2 ) ) ) ).

% dual_order.order_iff_strict
tff(fact_8090_dual__order_Ostrict__iff__order,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8091_dual__order_Ostrict__iff__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ord_less_set_nat(B2,A)
    <=> ( ord_less_eq_set_nat(B2,A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8092_dual__order_Ostrict__iff__order,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8093_dual__order_Ostrict__iff__order,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8094_dual__order_Ostrict__iff__order,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8095_dual__order_Ostrict__iff__order,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ( A != B2 ) ) ) ).

% dual_order.strict_iff_order
tff(fact_8096_dual__order_Ostrict__trans1,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_8097_dual__order_Ostrict__trans1,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ord_less_eq_set_nat(B2,A)
     => ( ord_less_set_nat(C,B2)
       => ord_less_set_nat(C,A) ) ) ).

% dual_order.strict_trans1
tff(fact_8098_dual__order_Ostrict__trans1,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_8099_dual__order_Ostrict__trans1,axiom,
    ! [B2: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,C),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_8100_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_8101_dual__order_Ostrict__trans1,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans1
tff(fact_8102_dual__order_Ostrict__trans2,axiom,
    ! [B2: real,A: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_8103_dual__order_Ostrict__trans2,axiom,
    ! [B2: set_nat,A: set_nat,C: set_nat] :
      ( ord_less_set_nat(B2,A)
     => ( ord_less_eq_set_nat(C,B2)
       => ord_less_set_nat(C,A) ) ) ).

% dual_order.strict_trans2
tff(fact_8104_dual__order_Ostrict__trans2,axiom,
    ! [B2: rat,A: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_8105_dual__order_Ostrict__trans2,axiom,
    ! [B2: num,A: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_8106_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_8107_dual__order_Ostrict__trans2,axiom,
    ! [B2: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% dual_order.strict_trans2
tff(fact_8108_dual__order_Ostrict__iff__not,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A)
        & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8109_dual__order_Ostrict__iff__not,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ord_less_set_nat(B2,A)
    <=> ( ord_less_eq_set_nat(B2,A)
        & ~ ord_less_eq_set_nat(A,B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8110_dual__order_Ostrict__iff__not,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A)
        & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8111_dual__order_Ostrict__iff__not,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A)
        & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8112_dual__order_Ostrict__iff__not,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A)
        & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8113_dual__order_Ostrict__iff__not,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A)
        & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% dual_order.strict_iff_not
tff(fact_8114_order_Ostrict__implies__order,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% order.strict_implies_order
tff(fact_8115_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_set_nat(A,B2)
     => ord_less_eq_set_nat(A,B2) ) ).

% order.strict_implies_order
tff(fact_8116_order_Ostrict__implies__order,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% order.strict_implies_order
tff(fact_8117_order_Ostrict__implies__order,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% order.strict_implies_order
tff(fact_8118_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% order.strict_implies_order
tff(fact_8119_order_Ostrict__implies__order,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% order.strict_implies_order
tff(fact_8120_dual__order_Ostrict__implies__order,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_8121_dual__order_Ostrict__implies__order,axiom,
    ! [B2: set_nat,A: set_nat] :
      ( ord_less_set_nat(B2,A)
     => ord_less_eq_set_nat(B2,A) ) ).

% dual_order.strict_implies_order
tff(fact_8122_dual__order_Ostrict__implies__order,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_8123_dual__order_Ostrict__implies__order,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_8124_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_8125_dual__order_Ostrict__implies__order,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ).

% dual_order.strict_implies_order
tff(fact_8126_order__le__less,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8127_order__le__less,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
    <=> ( ord_less_set_nat(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8128_order__le__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8129_order__le__less,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8130_order__le__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8131_order__le__less,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_less
tff(fact_8132_order__less__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8133_order__less__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_set_nat(Xb,Y)
    <=> ( ord_less_eq_set_nat(Xb,Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8134_order__less__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8135_order__less__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8136_order__less__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8137_order__less__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
        & ( Xb != Y ) ) ) ).

% order_less_le
tff(fact_8138_linorder__not__le,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_not_le
tff(fact_8139_linorder__not__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_not_le
tff(fact_8140_linorder__not__le,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_not_le
tff(fact_8141_linorder__not__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_not_le
tff(fact_8142_linorder__not__le,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_not_le
tff(fact_8143_linorder__not__less,axiom,
    ! [Xb: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Xb) ) ).

% linorder_not_less
tff(fact_8144_linorder__not__less,axiom,
    ! [Xb: rat,Y: rat] :
      ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Xb) ) ).

% linorder_not_less
tff(fact_8145_linorder__not__less,axiom,
    ! [Xb: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
    <=> aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Xb) ) ).

% linorder_not_less
tff(fact_8146_linorder__not__less,axiom,
    ! [Xb: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Xb) ) ).

% linorder_not_less
tff(fact_8147_linorder__not__less,axiom,
    ! [Xb: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Xb) ) ).

% linorder_not_less
tff(fact_8148_order__less__imp__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ).

% order_less_imp_le
tff(fact_8149_order__less__imp__le,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_set_nat(Xb,Y)
     => ord_less_eq_set_nat(Xb,Y) ) ).

% order_less_imp_le
tff(fact_8150_order__less__imp__le,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_8151_order__less__imp__le,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y) ) ).

% order_less_imp_le
tff(fact_8152_order__less__imp__le,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y) ) ).

% order_less_imp_le
tff(fact_8153_order__less__imp__le,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y) ) ).

% order_less_imp_le
tff(fact_8154_order__le__neq__trans,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ( A != B2 )
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_8155_order__le__neq__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ord_less_eq_set_nat(A,B2)
     => ( ( A != B2 )
       => ord_less_set_nat(A,B2) ) ) ).

% order_le_neq_trans
tff(fact_8156_order__le__neq__trans,axiom,
    ! [A: rat,B2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( ( A != B2 )
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_8157_order__le__neq__trans,axiom,
    ! [A: num,B2: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( ( A != B2 )
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_8158_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( ( A != B2 )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_8159_order__le__neq__trans,axiom,
    ! [A: int,B2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
     => ( ( A != B2 )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% order_le_neq_trans
tff(fact_8160_order__neq__le__trans,axiom,
    ! [A: real,B2: real] :
      ( ( A != B2 )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_8161_order__neq__le__trans,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( A != B2 )
     => ( ord_less_eq_set_nat(A,B2)
       => ord_less_set_nat(A,B2) ) ) ).

% order_neq_le_trans
tff(fact_8162_order__neq__le__trans,axiom,
    ! [A: rat,B2: rat] :
      ( ( A != B2 )
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_8163_order__neq__le__trans,axiom,
    ! [A: num,B2: num] :
      ( ( A != B2 )
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_8164_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_8165_order__neq__le__trans,axiom,
    ! [A: int,B2: int] :
      ( ( A != B2 )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% order_neq_le_trans
tff(fact_8166_order__le__less__trans,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Zb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Zb) ) ) ).

% order_le_less_trans
tff(fact_8167_order__le__less__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Zb: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ( ord_less_set_nat(Y,Zb)
       => ord_less_set_nat(Xb,Zb) ) ) ).

% order_le_less_trans
tff(fact_8168_order__le__less__trans,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Zb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Zb) ) ) ).

% order_le_less_trans
tff(fact_8169_order__le__less__trans,axiom,
    ! [Xb: num,Y: num,Zb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Zb)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Zb) ) ) ).

% order_le_less_trans
tff(fact_8170_order__le__less__trans,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Zb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Zb) ) ) ).

% order_le_less_trans
tff(fact_8171_order__le__less__trans,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Zb) ) ) ).

% order_le_less_trans
tff(fact_8172_order__less__le__trans,axiom,
    ! [Xb: real,Y: real,Zb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y),Zb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Zb) ) ) ).

% order_less_le_trans
tff(fact_8173_order__less__le__trans,axiom,
    ! [Xb: set_nat,Y: set_nat,Zb: set_nat] :
      ( ord_less_set_nat(Xb,Y)
     => ( ord_less_eq_set_nat(Y,Zb)
       => ord_less_set_nat(Xb,Zb) ) ) ).

% order_less_le_trans
tff(fact_8174_order__less__le__trans,axiom,
    ! [Xb: rat,Y: rat,Zb: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Y),Zb)
       => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Zb) ) ) ).

% order_less_le_trans
tff(fact_8175_order__less__le__trans,axiom,
    ! [Xb: num,Y: num,Zb: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Y),Zb)
       => aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Zb) ) ) ).

% order_less_le_trans
tff(fact_8176_order__less__le__trans,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Zb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Zb) ) ) ).

% order_less_le_trans
tff(fact_8177_order__less__le__trans,axiom,
    ! [Xb: int,Y: int,Zb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Zb)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Zb) ) ) ).

% order_less_le_trans
tff(fact_8178_order__le__less__subst1,axiom,
    ! [A: real,F: fun_real_real,B2: real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_real_real(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_8179_order__le__less__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_8180_order__le__less__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),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_8181_order__le__less__subst1,axiom,
    ! [A: real,F: fun_nat_real,B2: nat,C: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_nat_real(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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_8182_order__le__less__subst1,axiom,
    ! [A: real,F: fun_int_real,B2: int,C: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),aa_int_real(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),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_8183_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_real_rat,B2: real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_real_rat(F,B2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),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_le_less_subst1
tff(fact_8184_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),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_le_less_subst1
tff(fact_8185_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),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_le_less_subst1
tff(fact_8186_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_nat_rat,B2: nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_nat_rat(F,B2))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),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_le_less_subst1
tff(fact_8187_order__le__less__subst1,axiom,
    ! [A: rat,F: fun_int_rat,B2: int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),aa_int_rat(F,B2))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_int_rat(F,C)) ) ) ) ).

% order_le_less_subst1
tff(fact_8188_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(F,B2)),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_8189_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(F,B2)),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_8190_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_rat_num(F,B2)),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_8191_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_nat,C: nat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_rat_nat(F,B2)),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_8192_order__le__less__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_int,C: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_rat_int(F,B2)),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_8193_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_num_real(F,B2)),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_8194_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_num_rat(F,B2)),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_8195_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,aa_num_num(F,B2)),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_8196_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_nat,C: nat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,B2)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_8197_order__le__less__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_int,C: int] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,B2)),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_num_int(F,A)),C) ) ) ) ).

% order_le_less_subst2
tff(fact_8198_order__less__le__subst1,axiom,
    ! [A: real,F: fun_rat_real,B2: rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_rat_real(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_8199_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_rat_rat,B2: rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_rat_rat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_8200_order__less__le__subst1,axiom,
    ! [A: num,F: fun_rat_num,B2: rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_rat_num(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_8201_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_rat_nat,B2: rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_rat_nat(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_8202_order__less__le__subst1,axiom,
    ! [A: int,F: fun_rat_int,B2: rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_rat_int(F,B2))
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,B2),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_8203_order__less__le__subst1,axiom,
    ! [A: real,F: fun_num_real,B2: num,C: num] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),aa_num_real(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_8204_order__less__le__subst1,axiom,
    ! [A: rat,F: fun_num_rat,B2: num,C: num] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),aa_num_rat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_8205_order__less__le__subst1,axiom,
    ! [A: num,F: fun_num_num,B2: num,C: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),aa_num_num(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),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_8206_order__less__le__subst1,axiom,
    ! [A: nat,F: fun_num_nat,B2: num,C: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(F,X3)),aa_num_nat(F,Y3)) )
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_num_nat(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_8207_order__less__le__subst1,axiom,
    ! [A: int,F: fun_num_int,B2: num,C: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,B2))
     => ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,B2),C)
       => ( ! [X3: num,Y3: num] :
              ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,X3),Y3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(F,X3)),aa_num_int(F,Y3)) )
         => aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_num_int(F,C)) ) ) ) ).

% order_less_le_subst1
tff(fact_8208_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,B2)),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_8209_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_real,C: real] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(F,B2)),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_8210_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_real,C: real] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_num_real(F,B2)),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_8211_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_real,C: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,B2)),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_8212_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_real,C: real] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,B2)),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_8213_order__less__le__subst2,axiom,
    ! [A: real,B2: real,F: fun_real_rat,C: rat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,B2)),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_le_subst2
tff(fact_8214_order__less__le__subst2,axiom,
    ! [A: rat,B2: rat,F: fun_rat_rat,C: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_rat_rat(F,B2)),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_le_subst2
tff(fact_8215_order__less__le__subst2,axiom,
    ! [A: num,B2: num,F: fun_num_rat,C: rat] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_num_rat(F,B2)),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,aa_num_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_8216_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: fun_nat_rat,C: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,B2)),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,aa_nat_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_8217_order__less__le__subst2,axiom,
    ! [A: int,B2: int,F: fun_int_rat,C: rat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,B2)),C)
       => ( ! [X3: int,Y3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Y3)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,X3)),aa_int_rat(F,Y3)) )
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(F,A)),C) ) ) ) ).

% order_less_le_subst2
tff(fact_8218_linorder__le__less__linear,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
      | aa_real_o(aa_real_fun_real_o(ord_less_real,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_8219_linorder__le__less__linear,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
      | aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_8220_linorder__le__less__linear,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
      | aa_num_o(aa_num_fun_num_o(ord_less_num,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_8221_linorder__le__less__linear,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
      | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_8222_linorder__le__less__linear,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
      | aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Xb) ) ).

% linorder_le_less_linear
tff(fact_8223_order__le__imp__less__or__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8224_order__le__imp__less__or__eq,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ord_less_eq_set_nat(Xb,Y)
     => ( ord_less_set_nat(Xb,Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8225_order__le__imp__less__or__eq,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8226_order__le__imp__less__or__eq,axiom,
    ! [Xb: num,Y: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Xb),Y)
     => ( aa_num_o(aa_num_fun_num_o(ord_less_num,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8227_order__le__imp__less__or__eq,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8228_order__le__imp__less__or__eq,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Y)
        | ( Xb = Y ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_8229_and__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% and_not_numerals(8)
tff(fact_8230_not__int__rec,axiom,
    ! [K: int] : aa_int_int(bit_ri7919022796975470100ot_int,K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_o_int(zero_n2684676970156552555ol_int,aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% not_int_rec
tff(fact_8231__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
       => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,ma),na) = I4 )
           => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,ma),na)) )
          & ! [X4: nat] :
              ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),na) = I4 )
                & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),na)) )
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,mi),X4)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),ma) ) ) ) ) ) ).

% "5.hyps"(11)
tff(fact_8232__C20_C,axiom,
    ( ( za = mi )
    | ( za = ma )
    | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(treeList))
      & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ).

% "20"
tff(fact_8233_abd,axiom,
    ( ( za = ma )
   => aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% abd
tff(fact_8234_aaa,axiom,
    ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) )
   => aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% aaa
tff(fact_8235__C33_C,axiom,
    ~ ? [U2: nat] :
        ( aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),U2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),U2) ) ).

% "33"
tff(fact_8236__092_060open_062_092_060exists_062miny_O_Aboth__member__options_A_ItreeList_A_B_Asc_J_Aminy_092_060close_062,axiom,
    ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,sc)),X_1) ).

% \<open>\<exists>miny. both_member_options (treeList ! sc) miny\<close>
tff(fact_8237_abf,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% abf
tff(fact_8238_abe,axiom,
    aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% abe
tff(fact_8239_aa,axiom,
    ( ( za = ma )
   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% aa
tff(fact_8240_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I3) = nth_VEBT_VEBT(Ys2,I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_8241_nth__equalityI,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I3)
            <=> aa_nat_o(nth_o(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_8242_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I3) = aa_nat_nat(nth_nat(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_8243_nth__equalityI,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
     => ( ! [I3: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I3) = aa_nat_int(nth_int(Ys2),I3) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_8244_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_na6887940067898170868VEBT_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: vEBT_VEBT] : aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),X_13) )
    <=> ? [Xs3: list_VEBT_VEBT] :
          ( ( size_s6755466524823107622T_VEBT(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_VEBT_VEBT_o(aa_nat1925708948643195573VEBT_o(P,I2),nth_VEBT_VEBT(Xs3,I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_8245_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_o_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: $o] : aa_o_o(aa_nat_fun_o_o(P,I2),(X_13)) )
    <=> ? [Xs3: list_o] :
          ( ( size_size_list_o(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_o_o(aa_nat_fun_o_o(P,I2),aa_nat_o(nth_o(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_8246_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_nat_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: nat] : aa_nat_o(aa_nat_fun_nat_o(P,I2),X_13) )
    <=> ? [Xs3: list_nat] :
          ( ( size_size_list_nat(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_nat_o(aa_nat_fun_nat_o(P,I2),aa_nat_nat(nth_nat(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_8247_Skolem__list__nth,axiom,
    ! [K: nat,P: fun_nat_fun_int_o] :
      ( ! [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
         => ? [X_13: int] : aa_int_o(aa_nat_fun_int_o(P,I2),X_13) )
    <=> ? [Xs3: list_int] :
          ( ( size_size_list_int(Xs3) = K )
          & ! [I2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),K)
             => aa_int_o(aa_nat_fun_int_o(P,I2),aa_nat_int(nth_int(Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_8248_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
      ( ( Xs = Ys2 )
    <=> ( ( size_s6755466524823107622T_VEBT(Xs) = size_s6755466524823107622T_VEBT(Ys2) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
           => ( nth_VEBT_VEBT(Xs,I2) = nth_VEBT_VEBT(Ys2,I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_8249_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_o,Ys2: list_o] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_o(Xs) = size_size_list_o(Ys2) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Xs))
           => ( aa_nat_o(nth_o(Xs),I2)
            <=> aa_nat_o(nth_o(Ys2),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_8250_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys2) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Xs))
           => ( aa_nat_nat(nth_nat(Xs),I2) = aa_nat_nat(nth_nat(Ys2),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_8251_list__eq__iff__nth__eq,axiom,
    ! [Xs: list_int,Ys2: list_int] :
      ( ( Xs = Ys2 )
    <=> ( ( size_size_list_int(Xs) = size_size_list_int(Ys2) )
        & ! [I2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
           => ( aa_nat_int(nth_int(Xs),I2) = aa_nat_int(nth_int(Ys2),I2) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_8252_num_Osize_I4_J,axiom,
    aa_num_nat(size_size_num,one) = zero_zero_nat ).

% num.size(4)
tff(fact_8253_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_8254_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_8255_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list_VEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5917875025757280293ildren(Nb,TreeLista),Xb)
    <=> aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),Nb))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),Nb)) ) ).

% in_children_def
tff(fact_8256_abc,axiom,
    aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),na) ).

% abc
tff(fact_8257__092_060open_062invar__vebt_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),na) ).

% \<open>invar_vebt (treeList ! high x (deg div 2)) n\<close>
tff(fact_8258_abh,axiom,
    aa_nat_o(vEBT_vebt_member(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% abh
tff(fact_8259_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ aa_nat_o(vEBT_invar_vebt(Ta),zero_zero_nat) ).

% valid_0_not
tff(fact_8260_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_8261__C5_Ohyps_C_I2_J,axiom,
    aa_nat_o(vEBT_invar_vebt(summary),m) ).

% "5.hyps"(2)
tff(fact_8262_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_8263_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_8264_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_8265__092_060open_062both__member__options_Asummary_Asc_092_060close_062,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(summary),sc) ).

% \<open>both_member_options summary sc\<close>
tff(fact_8266__092_060open_062vebt__member_Asummary_Asc_092_060close_062,axiom,
    aa_nat_o(vEBT_vebt_member(summary),sc) ).

% \<open>vebt_member summary sc\<close>
tff(fact_8267_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
      <=> member_nat(Xb,vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_8268_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
     => ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% member_bound
tff(fact_8269_abg,axiom,
    aa_nat_o(vEBT_V8194947554948674370ptions(summary),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% abg
tff(fact_8270__092_060open_062length_AtreeList_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_Am_092_060close_062,axiom,
    ( ( size_s6755466524823107622T_VEBT(treeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m) )
    & aa_nat_o(vEBT_invar_vebt(summary),m) ) ).

% \<open>length treeList = 2 ^ m \<and> invar_vebt summary m\<close>
tff(fact_8271__C5_Ohyps_C_I7_J,axiom,
    ! [I4: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m))
     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(treeList,I4)),X_13)
      <=> aa_nat_o(vEBT_V8194947554948674370ptions(summary),I4) ) ) ).

% "5.hyps"(7)
tff(fact_8272_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa_nat_o(vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_8273_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
         => ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_8274_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))
       => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_8275_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(Info,Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(Info,Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_8276_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_8277_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),aa_nat_nat(suc,aa_nat_nat(suc,Nb)))
     => ? [Info2: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa_nat_nat(suc,aa_nat_nat(suc,Nb)),TreeList,S3) ) ).

% deg_SUcn_Node
tff(fact_8278__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X4: vEBT_VEBT] :
        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(treeList))
       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) ) ).

% "5.hyps"(8)
tff(fact_8279_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_8280_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_8281__092_060open_062is__succ__in__set_A_Iset__vebt_H_Asummary_J_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_Asc_092_060close_062,axiom,
    aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(summary)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),sc) ).

% \<open>is_succ_in_set (set_vebt' summary) (high x (deg div 2)) sc\<close>
tff(fact_8282_set__n__deg__not__0,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Mb: 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))),Mb) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_8283_inthall,axiom,
    ! [Xs: list_Extended_enat,P: fun_Extended_enat_o,Nb: nat] :
      ( ! [X3: extended_enat] :
          ( member_Extended_enat(X3,set_Extended_enat2(Xs))
         => aa_Extended_enat_o(P,X3) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3941691890525107288d_enat(Xs))
       => aa_Extended_enat_o(P,nth_Extended_enat(Xs,Nb)) ) ) ).

% inthall
tff(fact_8284_inthall,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Nb: nat] :
      ( ! [X3: complex] :
          ( 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_8285_inthall,axiom,
    ! [Xs: list_real,P: fun_real_o,Nb: nat] :
      ( ! [X3: real] :
          ( 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_8286_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_8287_inthall,axiom,
    ! [Xs: list_o,P: fun_o_o,Nb: nat] :
      ( ! [X3: $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_8288_inthall,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Nb: nat] :
      ( ! [X3: nat] :
          ( 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_8289_inthall,axiom,
    ! [Xs: list_int,P: fun_int_o,Nb: nat] :
      ( ! [X3: int] :
          ( 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_8290_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta)),Xb),Y)
    <=> ( aa_nat_o(vEBT_vebt_member(Ta),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [Z4: nat] :
            ( ( aa_nat_o(vEBT_vebt_member(Ta),Z4)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Z4) )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z4) ) ) ) ).

% succ_member
tff(fact_8291_length__pos__if__in__set,axiom,
    ! [Xb: extended_enat,Xs: list_Extended_enat] :
      ( member_Extended_enat(Xb,set_Extended_enat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3941691890525107288d_enat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8292_length__pos__if__in__set,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s3451745648224563538omplex(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8293_length__pos__if__in__set,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_real(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8294_length__pos__if__in__set,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6755466524823107622T_VEBT(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8295_length__pos__if__in__set,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_o(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8296_length__pos__if__in__set,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( member_nat(Xb,set_nat2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8297_length__pos__if__in__set,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs)) ) ).

% length_pos_if_in_set
tff(fact_8298_nth__mem,axiom,
    ! [Nb: nat,Xs: list_Extended_enat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3941691890525107288d_enat(Xs))
     => member_Extended_enat(nth_Extended_enat(Xs,Nb),set_Extended_enat2(Xs)) ) ).

% nth_mem
tff(fact_8299_nth__mem,axiom,
    ! [Nb: nat,Xs: list_complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_s3451745648224563538omplex(Xs))
     => member_complex(aa_nat_complex(nth_complex(Xs),Nb),set_complex2(Xs)) ) ).

% nth_mem
tff(fact_8300_nth__mem,axiom,
    ! [Nb: nat,Xs: list_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_real(Xs))
     => member_real(aa_nat_real(nth_real(Xs),Nb),set_real2(Xs)) ) ).

% nth_mem
tff(fact_8301_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_8302_nth__mem,axiom,
    ! [Nb: nat,Xs: list_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_o(Xs))
     => member_o(aa_nat_o(nth_o(Xs),Nb),set_o2(Xs)) ) ).

% nth_mem
tff(fact_8303_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))
     => member_nat(aa_nat_nat(nth_nat(Xs),Nb),set_nat2(Xs)) ) ).

% nth_mem
tff(fact_8304_nth__mem,axiom,
    ! [Nb: nat,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),size_size_list_int(Xs))
     => member_int(aa_nat_int(nth_int(Xs),Nb),set_int2(Xs)) ) ).

% nth_mem
tff(fact_8305_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_8306_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] :
            ( 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_8307_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] :
            ( 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_8308_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] :
            ( 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_8309_in__set__conv__nth,axiom,
    ! [Xb: extended_enat,Xs: list_Extended_enat] :
      ( member_Extended_enat(Xb,set_Extended_enat2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3941691890525107288d_enat(Xs))
          & ( nth_Extended_enat(Xs,I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8310_in__set__conv__nth,axiom,
    ! [Xb: complex,Xs: list_complex] :
      ( member_complex(Xb,set_complex2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Xs))
          & ( aa_nat_complex(nth_complex(Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8311_in__set__conv__nth,axiom,
    ! [Xb: real,Xs: list_real] :
      ( member_real(Xb,set_real2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Xs))
          & ( aa_nat_real(nth_real(Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8312_in__set__conv__nth,axiom,
    ! [Xb: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
    <=> ? [I2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Xs))
          & ( nth_VEBT_VEBT(Xs,I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8313_in__set__conv__nth,axiom,
    ! [Xb: $o,Xs: list_o] :
      ( member_o((Xb),set_o2(Xs))
    <=> ? [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)
          <=> (Xb) ) ) ) ).

% in_set_conv_nth
tff(fact_8314_in__set__conv__nth,axiom,
    ! [Xb: nat,Xs: list_nat] :
      ( member_nat(Xb,set_nat2(Xs))
    <=> ? [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) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8315_in__set__conv__nth,axiom,
    ! [Xb: int,Xs: list_int] :
      ( member_int(Xb,set_int2(Xs))
    <=> ? [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) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_8316_all__nth__imp__all__set,axiom,
    ! [Xs: list_Extended_enat,P: fun_Extended_enat_o,Xb: extended_enat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3941691890525107288d_enat(Xs))
         => aa_Extended_enat_o(P,nth_Extended_enat(Xs,I3)) )
     => ( member_Extended_enat(Xb,set_Extended_enat2(Xs))
       => aa_Extended_enat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8317_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: fun_complex_o,Xb: complex] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s3451745648224563538omplex(Xs))
         => aa_complex_o(P,aa_nat_complex(nth_complex(Xs),I3)) )
     => ( member_complex(Xb,set_complex2(Xs))
       => aa_complex_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8318_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: fun_real_o,Xb: real] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_real(Xs))
         => aa_real_o(P,aa_nat_real(nth_real(Xs),I3)) )
     => ( member_real(Xb,set_real2(Xs))
       => aa_real_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8319_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o,Xb: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_s6755466524823107622T_VEBT(Xs))
         => aa_VEBT_VEBT_o(P,nth_VEBT_VEBT(Xs,I3)) )
     => ( member_VEBT_VEBT(Xb,set_VEBT_VEBT2(Xs))
       => aa_VEBT_VEBT_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8320_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: fun_o_o,Xb: $o] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_o(Xs))
         => aa_o_o(P,aa_nat_o(nth_o(Xs),I3)) )
     => ( member_o((Xb),set_o2(Xs))
       => aa_o_o(P,(Xb)) ) ) ).

% all_nth_imp_all_set
tff(fact_8321_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: fun_nat_o,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_nat(Xs))
         => aa_nat_o(P,aa_nat_nat(nth_nat(Xs),I3)) )
     => ( member_nat(Xb,set_nat2(Xs))
       => aa_nat_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8322_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: fun_int_o,Xb: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),size_size_list_int(Xs))
         => aa_int_o(P,aa_nat_int(nth_int(Xs),I3)) )
     => ( member_int(Xb,set_int2(Xs))
       => aa_int_o(P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_8323_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: fun_VEBT_VEBT_o] :
      ( ! [X: vEBT_VEBT] :
          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Xs))
         => aa_VEBT_VEBT_o(P,X) )
    <=> ! [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)) ) ) ).

% all_set_conv_all_nth
tff(fact_8324_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: fun_o_o] :
      ( ! [X: $o] :
          ( member_o((X),set_o2(Xs))
         => aa_o_o(P,(X)) )
    <=> ! [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)) ) ) ).

% all_set_conv_all_nth
tff(fact_8325_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: fun_nat_o] :
      ( ! [X: nat] :
          ( member_nat(X,set_nat2(Xs))
         => aa_nat_o(P,X) )
    <=> ! [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)) ) ) ).

% all_set_conv_all_nth
tff(fact_8326_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: fun_int_o] :
      ( ! [X: int] :
          ( member_int(X,set_int2(Xs))
         => aa_int_o(P,X) )
    <=> ! [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)) ) ) ).

% all_set_conv_all_nth
tff(fact_8327_is__succ__in__set__def,axiom,
    ! [Xs: set_nat,Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),Xb),Y)
    <=> ( member_nat(Y,Xs)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
        & ! [X: nat] :
            ( member_nat(X,Xs)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),X)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_8328_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => 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_8329_fgh,axiom,
    vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,sc)) != bot_bot_set_nat ).

% fgh
tff(fact_8330__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Az_A_092_060and_062_Ax_A_060_Az_092_060close_062,axiom,
    ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),za)
    & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,xa),za) ) ).

% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) z \<and> x < z\<close>
tff(fact_8331__092_060open_062vebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_092_060close_062,axiom,
    vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = some_nat(sc) ).

% \<open>vebt_succ summary (high x (deg div 2)) = Some sc\<close>
tff(fact_8332_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot_set_nat ).

% buildup_gives_empty
tff(fact_8333_power__shift,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Xb),Y) = Zb )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_power,some_nat(Xb)),some_nat(Y)) = some_nat(Zb) ) ) ).

% power_shift
tff(fact_8334__C5_Ohyps_C_I1_J,axiom,
    ! [X4: vEBT_VEBT] :
      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(treeList))
     => ( aa_nat_o(vEBT_invar_vebt(X4),na)
        & ! [Xa: nat,Xb2: nat] :
            ( ( vEBT_vebt_succ(X4,Xa) = some_nat(Xb2) )
          <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(X4)),Xa),Xb2) ) ) ) ).

% "5.hyps"(1)
tff(fact_8335_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia = Maa )
       => ( ! [X4: vEBT_VEBT] :
              ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeLista))
             => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) )
          & ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_8336__C5_Ohyps_C_I3_J,axiom,
    ! [Xb: nat,Sxb: nat] :
      ( ( vEBT_vebt_succ(summary,Xb) = some_nat(Sxb) )
    <=> aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(summary)),Xb),Sxb) ) ).

% "5.hyps"(3)
tff(fact_8337_atLeastAtMost__iff,axiom,
    ! [I: extended_enat,L: extended_enat,U: extended_enat] :
      ( member_Extended_enat(I,set_or5403411693681687835d_enat(L,U))
    <=> ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(L),I)
        & aa_Extended_enat_o(ord_le2932123472753598470d_enat(I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8338_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( member_set_nat(I,set_or4548717258645045905et_nat(L,U))
    <=> ( ord_less_eq_set_nat(L,I)
        & ord_less_eq_set_nat(I,U) ) ) ).

% atLeastAtMost_iff
tff(fact_8339_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( member_rat(I,set_or633870826150836451st_rat(L,U))
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),I)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8340_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( member_num(I,set_or7049704709247886629st_num(L,U))
    <=> ( aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),I)
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8341_atLeastAtMost__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( member_nat(I,aa_nat_set_nat(set_or1269000886237332187st_nat(L),U))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),I)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8342_atLeastAtMost__iff,axiom,
    ! [I: int,L: int,U: int] :
      ( member_int(I,set_or1266510415728281911st_int(L,U))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),I)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8343_atLeastAtMost__iff,axiom,
    ! [I: real,L: real,U: real] :
      ( member_real(I,set_or1222579329274155063t_real(L,U))
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),I)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,I),U) ) ) ).

% atLeastAtMost_iff
tff(fact_8344_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L3: set_nat,H2: set_nat] :
      ( ( set_or4548717258645045905et_nat(L,H) = set_or4548717258645045905et_nat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ ord_less_eq_set_nat(L,H)
          & ~ ord_less_eq_set_nat(L3,H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8345_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H2: rat] :
      ( ( set_or633870826150836451st_rat(L,H) = set_or633870826150836451st_rat(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L),H)
          & ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8346_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H2: num] :
      ( ( set_or7049704709247886629st_num(L,H) = set_or7049704709247886629st_num(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L),H)
          & ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8347_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(L),H) = aa_nat_set_nat(set_or1269000886237332187st_nat(L3),H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L),H)
          & ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8348_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H2: int] :
      ( ( set_or1266510415728281911st_int(L,H) = set_or1266510415728281911st_int(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L),H)
          & ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8349_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H2: real] :
      ( ( set_or1222579329274155063t_real(L,H) = set_or1222579329274155063t_real(L3,H2) )
    <=> ( ( ( L = L3 )
          & ( H = H2 ) )
        | ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L),H)
          & ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L3),H2) ) ) ) ).

% Icc_eq_Icc
tff(fact_8350_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
       => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_8351_Diff__cancel,axiom,
    ! [A3: set_Extended_enat] : minus_925952699566721837d_enat(A3,A3) = bot_bo7653980558646680370d_enat ).

% Diff_cancel
tff(fact_8352_Diff__cancel,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,A3) = bot_bot_set_real ).

% Diff_cancel
tff(fact_8353_Diff__cancel,axiom,
    ! [A3: set_int] : minus_minus_set_int(A3,A3) = bot_bot_set_int ).

% Diff_cancel
tff(fact_8354_Diff__cancel,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(A3,A3) = bot_bot_set_nat ).

% Diff_cancel
tff(fact_8355_empty__Diff,axiom,
    ! [A3: set_Extended_enat] : minus_925952699566721837d_enat(bot_bo7653980558646680370d_enat,A3) = bot_bo7653980558646680370d_enat ).

% empty_Diff
tff(fact_8356_empty__Diff,axiom,
    ! [A3: set_real] : minus_minus_set_real(bot_bot_set_real,A3) = bot_bot_set_real ).

% empty_Diff
tff(fact_8357_empty__Diff,axiom,
    ! [A3: set_int] : minus_minus_set_int(bot_bot_set_int,A3) = bot_bot_set_int ).

% empty_Diff
tff(fact_8358_empty__Diff,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(bot_bot_set_nat,A3) = bot_bot_set_nat ).

% empty_Diff
tff(fact_8359_Diff__empty,axiom,
    ! [A3: set_Extended_enat] : minus_925952699566721837d_enat(A3,bot_bo7653980558646680370d_enat) = A3 ).

% Diff_empty
tff(fact_8360_Diff__empty,axiom,
    ! [A3: set_real] : minus_minus_set_real(A3,bot_bot_set_real) = A3 ).

% Diff_empty
tff(fact_8361_Diff__empty,axiom,
    ! [A3: set_int] : minus_minus_set_int(A3,bot_bot_set_int) = A3 ).

% Diff_empty
tff(fact_8362_Diff__empty,axiom,
    ! [A3: set_nat] : minus_minus_set_nat(A3,bot_bot_set_nat) = A3 ).

% Diff_empty
tff(fact_8363_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)) ) ) ).

% mi_ma_2_deg
tff(fact_8364_succ__min,axiom,
    ! [Dega: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = some_nat(Mia) ) ) ) ).

% succ_min
tff(fact_8365_atLeastatMost__empty__iff,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat )
    <=> ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8366_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( set_or4548717258645045905et_nat(A,B2) = bot_bot_set_set_nat )
    <=> ~ ord_less_eq_set_nat(A,B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8367_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B2: rat] :
      ( ( set_or633870826150836451st_rat(A,B2) = bot_bot_set_rat )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8368_atLeastatMost__empty__iff,axiom,
    ! [A: num,B2: num] :
      ( ( set_or7049704709247886629st_num(A,B2) = bot_bot_set_num )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8369_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = bot_bot_set_nat )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8370_atLeastatMost__empty__iff,axiom,
    ! [A: int,B2: int] :
      ( ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8371_atLeastatMost__empty__iff,axiom,
    ! [A: real,B2: real] :
      ( ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% atLeastatMost_empty_iff
tff(fact_8372_atLeastatMost__empty__iff2,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( bot_bo7653980558646680370d_enat = set_or5403411693681687835d_enat(A,B2) )
    <=> ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8373_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B2: set_nat] :
      ( ( bot_bot_set_set_nat = set_or4548717258645045905et_nat(A,B2) )
    <=> ~ ord_less_eq_set_nat(A,B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8374_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B2: rat] :
      ( ( bot_bot_set_rat = set_or633870826150836451st_rat(A,B2) )
    <=> ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8375_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B2: num] :
      ( ( bot_bot_set_num = set_or7049704709247886629st_num(A,B2) )
    <=> ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8376_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B2: nat] :
      ( ( bot_bot_set_nat = aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) )
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8377_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B2: int] :
      ( ( bot_bot_set_int = set_or1266510415728281911st_int(A,B2) )
    <=> ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8378_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B2: real] :
      ( ( bot_bot_set_real = set_or1222579329274155063t_real(A,B2) )
    <=> ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2) ) ).

% atLeastatMost_empty_iff2
tff(fact_8379_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ord_le6893508408891458716et_nat(set_or4548717258645045905et_nat(A,B2),set_or4548717258645045905et_nat(C,D))
    <=> ( ~ ord_less_eq_set_nat(A,B2)
        | ( ord_less_eq_set_nat(C,A)
          & ord_less_eq_set_nat(B2,D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8380_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ord_less_eq_set_rat(set_or633870826150836451st_rat(A,B2),set_or633870826150836451st_rat(C,D))
    <=> ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
        | ( 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,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8381_atLeastatMost__subset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ord_less_eq_set_num(set_or7049704709247886629st_num(A,B2),set_or7049704709247886629st_num(C,D))
    <=> ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
        | ( 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,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8382_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ord_less_eq_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
        | ( 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,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8383_atLeastatMost__subset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ord_less_eq_set_int(set_or1266510415728281911st_int(A,B2),set_or1266510415728281911st_int(C,D))
    <=> ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
        | ( 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,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8384_atLeastatMost__subset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ord_less_eq_set_real(set_or1222579329274155063t_real(A,B2),set_or1222579329274155063t_real(C,D))
    <=> ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
        | ( 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,B2),D) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_8385_atLeastatMost__empty,axiom,
    ! [B2: extended_enat,A: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(B2),A)
     => ( set_or5403411693681687835d_enat(A,B2) = bot_bo7653980558646680370d_enat ) ) ).

% atLeastatMost_empty
tff(fact_8386_atLeastatMost__empty,axiom,
    ! [B2: rat,A: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B2),A)
     => ( set_or633870826150836451st_rat(A,B2) = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
tff(fact_8387_atLeastatMost__empty,axiom,
    ! [B2: num,A: num] :
      ( aa_num_o(aa_num_fun_num_o(ord_less_num,B2),A)
     => ( set_or7049704709247886629st_num(A,B2) = bot_bot_set_num ) ) ).

% atLeastatMost_empty
tff(fact_8388_atLeastatMost__empty,axiom,
    ! [B2: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B2),A)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2) = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
tff(fact_8389_atLeastatMost__empty,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A)
     => ( set_or1266510415728281911st_int(A,B2) = bot_bot_set_int ) ) ).

% atLeastatMost_empty
tff(fact_8390_atLeastatMost__empty,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => ( set_or1222579329274155063t_real(A,B2) = bot_bot_set_real ) ) ).

% atLeastatMost_empty
tff(fact_8391_Diff__eq__empty__iff,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(A3,B3) = bot_bo7653980558646680370d_enat )
    <=> ord_le7203529160286727270d_enat(A3,B3) ) ).

% Diff_eq_empty_iff
tff(fact_8392_Diff__eq__empty__iff,axiom,
    ! [A3: set_real,B3: set_real] :
      ( ( minus_minus_set_real(A3,B3) = bot_bot_set_real )
    <=> ord_less_eq_set_real(A3,B3) ) ).

% Diff_eq_empty_iff
tff(fact_8393_Diff__eq__empty__iff,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( minus_minus_set_int(A3,B3) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A3,B3) ) ).

% Diff_eq_empty_iff
tff(fact_8394_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat(A3,B3) = bot_bot_set_nat )
    <=> ord_less_eq_set_nat(A3,B3) ) ).

% Diff_eq_empty_iff
tff(fact_8395__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062sc_O_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_061_ASome_Asc_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Sc: nat] : vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) != some_nat(Sc) ).

% \<open>\<And>thesis. (\<And>sc. vebt_succ summary (high x (deg div 2)) = Some sc \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_8396_both__member__options__from__complete__tree__to__child,axiom,
    ! [Dega: nat,Mia: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
     => ( aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_8397_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Maa)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
            & aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ) ) ) ).

% member_inv
tff(fact_8398_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Dega)
       => ( aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => aa_nat_o(vEBT_V8194947554948674370ptions(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_8399__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_092_060close_062,axiom,
    some_nat(miny) = vEBT_vebt_mint(nth_VEBT_VEBT(treeList,sc)) ).

% \<open>Some miny = vebt_mint (treeList ! sc)\<close>
tff(fact_8400_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Xb)
    <=> vEBT_VEBT_greater(some_nat(Xb),some_nat(Y)) ) ).

% greater_shift
tff(fact_8401_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Y)
    <=> vEBT_VEBT_lesseq(some_nat(Xb),some_nat(Y)) ) ).

% lesseq_shift
tff(fact_8402_False,axiom,
    vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) != none_nat ).

% False
tff(fact_8403_not__psubset__empty,axiom,
    ! [A3: set_Extended_enat] : ~ ord_le2529575680413868914d_enat(A3,bot_bo7653980558646680370d_enat) ).

% not_psubset_empty
tff(fact_8404_not__psubset__empty,axiom,
    ! [A3: set_real] : ~ ord_less_set_real(A3,bot_bot_set_real) ).

% not_psubset_empty
tff(fact_8405_not__psubset__empty,axiom,
    ! [A3: set_nat] : ~ ord_less_set_nat(A3,bot_bot_set_nat) ).

% not_psubset_empty
tff(fact_8406_not__psubset__empty,axiom,
    ! [A3: set_int] : ~ ord_less_set_int(A3,bot_bot_set_int) ).

% not_psubset_empty
tff(fact_8407_bot_Onot__eq__extremum,axiom,
    ! [A: set_Extended_enat] :
      ( ( A != bot_bo7653980558646680370d_enat )
    <=> ord_le2529575680413868914d_enat(bot_bo7653980558646680370d_enat,A) ) ).

% bot.not_eq_extremum
tff(fact_8408_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
    <=> ord_less_set_real(bot_bot_set_real,A) ) ).

% bot.not_eq_extremum
tff(fact_8409_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
    <=> ord_less_set_nat(bot_bot_set_nat,A) ) ).

% bot.not_eq_extremum
tff(fact_8410_bot_Onot__eq__extremum,axiom,
    ! [A: set_int] :
      ( ( A != bot_bot_set_int )
    <=> ord_less_set_int(bot_bot_set_int,A) ) ).

% bot.not_eq_extremum
tff(fact_8411_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_8412_bot_Oextremum__strict,axiom,
    ! [A: set_Extended_enat] : ~ ord_le2529575680413868914d_enat(A,bot_bo7653980558646680370d_enat) ).

% bot.extremum_strict
tff(fact_8413_bot_Oextremum__strict,axiom,
    ! [A: set_real] : ~ ord_less_set_real(A,bot_bot_set_real) ).

% bot.extremum_strict
tff(fact_8414_bot_Oextremum__strict,axiom,
    ! [A: set_nat] : ~ ord_less_set_nat(A,bot_bot_set_nat) ).

% bot.extremum_strict
tff(fact_8415_bot_Oextremum__strict,axiom,
    ! [A: set_int] : ~ ord_less_set_int(A,bot_bot_set_int) ).

% bot.extremum_strict
tff(fact_8416_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_8417_bot_Oextremum,axiom,
    ! [A: set_Extended_enat] : ord_le7203529160286727270d_enat(bot_bo7653980558646680370d_enat,A) ).

% bot.extremum
tff(fact_8418_bot_Oextremum,axiom,
    ! [A: set_real] : ord_less_eq_set_real(bot_bot_set_real,A) ).

% bot.extremum
tff(fact_8419_bot_Oextremum,axiom,
    ! [A: set_int] : ord_less_eq_set_int(bot_bot_set_int,A) ).

% bot.extremum
tff(fact_8420_bot_Oextremum,axiom,
    ! [A: set_nat] : ord_less_eq_set_nat(bot_bot_set_nat,A) ).

% bot.extremum
tff(fact_8421_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_8422_bot_Oextremum__unique,axiom,
    ! [A: set_Extended_enat] :
      ( ord_le7203529160286727270d_enat(A,bot_bo7653980558646680370d_enat)
    <=> ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_unique
tff(fact_8423_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
    <=> ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
tff(fact_8424_bot_Oextremum__unique,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
    <=> ( A = bot_bot_set_int ) ) ).

% bot.extremum_unique
tff(fact_8425_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ord_less_eq_set_nat(A,bot_bot_set_nat)
    <=> ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
tff(fact_8426_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_8427_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Extended_enat] :
      ( ord_le7203529160286727270d_enat(A,bot_bo7653980558646680370d_enat)
     => ( A = bot_bo7653980558646680370d_enat ) ) ).

% bot.extremum_uniqueI
tff(fact_8428_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ord_less_eq_set_real(A,bot_bot_set_real)
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
tff(fact_8429_bot_Oextremum__uniqueI,axiom,
    ! [A: set_int] :
      ( ord_less_eq_set_int(A,bot_bot_set_int)
     => ( A = bot_bot_set_int ) ) ).

% bot.extremum_uniqueI
tff(fact_8430_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ord_less_eq_set_nat(A,bot_bot_set_nat)
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
tff(fact_8431_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_8432_diff__shunt__var,axiom,
    ! [Xb: set_Extended_enat,Y: set_Extended_enat] :
      ( ( minus_925952699566721837d_enat(Xb,Y) = bot_bo7653980558646680370d_enat )
    <=> ord_le7203529160286727270d_enat(Xb,Y) ) ).

% diff_shunt_var
tff(fact_8433_diff__shunt__var,axiom,
    ! [Xb: set_real,Y: set_real] :
      ( ( minus_minus_set_real(Xb,Y) = bot_bot_set_real )
    <=> ord_less_eq_set_real(Xb,Y) ) ).

% diff_shunt_var
tff(fact_8434_diff__shunt__var,axiom,
    ! [Xb: set_int,Y: set_int] :
      ( ( minus_minus_set_int(Xb,Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(Xb,Y) ) ).

% diff_shunt_var
tff(fact_8435_diff__shunt__var,axiom,
    ! [Xb: set_nat,Y: set_nat] :
      ( ( minus_minus_set_nat(Xb,Y) = bot_bot_set_nat )
    <=> ord_less_eq_set_nat(Xb,Y) ) ).

% diff_shunt_var
tff(fact_8436_ex__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),Nb)
          & aa_nat_o(P,M6) )
    <=> ? [X: nat] :
          ( member_nat(X,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less
tff(fact_8437_all__nat__less,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),Nb)
         => aa_nat_o(P,M6) )
    <=> ! [X: nat] :
          ( member_nat(X,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less
tff(fact_8438_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B2: set_nat,C: set_nat,D: set_nat] :
      ( ord_less_set_set_nat(set_or4548717258645045905et_nat(A,B2),set_or4548717258645045905et_nat(C,D))
    <=> ( ( ~ ord_less_eq_set_nat(A,B2)
          | ( ord_less_eq_set_nat(C,A)
            & ord_less_eq_set_nat(B2,D)
            & ( ord_less_set_nat(C,A)
              | ord_less_set_nat(B2,D) ) ) )
        & ord_less_eq_set_nat(C,D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8439_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B2: rat,C: rat,D: rat] :
      ( ord_less_set_rat(set_or633870826150836451st_rat(A,B2),set_or633870826150836451st_rat(C,D))
    <=> ( ( ~ aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,A),B2)
          | ( 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,B2),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,B2),D) ) ) )
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8440_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B2: num,C: num,D: num] :
      ( ord_less_set_num(set_or7049704709247886629st_num(A,B2),set_or7049704709247886629st_num(C,D))
    <=> ( ( ~ aa_num_o(aa_num_fun_num_o(ord_less_eq_num,A),B2)
          | ( 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,B2),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,B2),D) ) ) )
        & aa_num_o(aa_num_fun_num_o(ord_less_eq_num,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8441_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B2: nat,C: nat,D: nat] :
      ( ord_less_set_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2),aa_nat_set_nat(set_or1269000886237332187st_nat(C),D))
    <=> ( ( ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
          | ( 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,B2),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,B2),D) ) ) )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8442_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B2: int,C: int,D: int] :
      ( ord_less_set_int(set_or1266510415728281911st_int(A,B2),set_or1266510415728281911st_int(C,D))
    <=> ( ( ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2)
          | ( 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,B2),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,B2),D) ) ) )
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8443_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B2: real,C: real,D: real] :
      ( ord_less_set_real(set_or1222579329274155063t_real(A,B2),set_or1222579329274155063t_real(C,D))
    <=> ( ( ~ aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
          | ( 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,B2),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,B2),D) ) ) )
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C),D) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_8444_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: 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),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),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) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),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(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_8445_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: nat,Dega: nat,Mia: 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),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ! [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))),Mb))
                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),X_13)
                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X3: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                       => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Maa)
                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega))
                     => ( ( ( Mia != Maa )
                         => ! [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))),Mb))
                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),Nb) = I3 )
                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),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) = I3 )
                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeLista,I3)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X3),Nb)) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),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(Mia),Maa)),Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_8446_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Nb)
     => ( ( Mia != Maa )
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Maa)
          & ? [M3: nat] :
              ( ( some_nat(M3) = vEBT_vebt_mint(Summarya) )
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,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_8447__092_060open_062min__in__set_A_Iset__vebt_H_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_Aminy_092_060close_062,axiom,
    aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))),miny) ).

% \<open>min_in_set (set_vebt' (treeList ! the (vebt_succ summary (high x (deg div 2))))) miny\<close>
tff(fact_8448_scmem,axiom,
    aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),miny) ).

% scmem
tff(fact_8449__092_060open_062invar__vebt_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_An_092_060close_062,axiom,
    aa_nat_o(vEBT_invar_vebt(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))),na) ).

% \<open>invar_vebt (treeList ! the (vebt_succ summary (high x (deg div 2)))) n\<close>
tff(fact_8450_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_8451_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Mini) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_8452_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_8453_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_mint(Ta) = some_nat(Xb) ) ) ) ).

% mint_sound
tff(fact_8454_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_mint(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% mint_corr
tff(fact_8455__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062miny_O_ASome_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Asc_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Miny: nat] : some_nat(Miny) != vEBT_vebt_mint(nth_VEBT_VEBT(treeList,sc)) ).

% \<open>\<And>thesis. (\<And>miny. Some miny = vebt_mint (treeList ! sc) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
tff(fact_8456_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( some_nat(Mb) = vEBT_vebt_mint(Ta) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ) ) ).

% misiz
tff(fact_8457_bit__0__eq,axiom,
    bit_se1146084159140164899it_int(zero_zero_int) = bot_bot_fun_nat_o ).

% bit_0_eq
tff(fact_8458_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_8459_succ__list__to__short,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = none_nat ) ) ) ) ).

% succ_list_to_short
tff(fact_8460_nested__mint,axiom,
    ! [Mia: nat,Maa: 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(Mia),Maa)),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,Maa),Mia)
         => ( ( Maa != Mia )
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,the_nat(vEBT_vebt_mint(Summarya))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,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_8461__092_060open_062Some_Aminy_A_061_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
    some_nat(miny) = vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% \<open>Some miny = vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
tff(fact_8462_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V4262088993061758097ft_nat(power_power_nat) ).

% local.power_def
tff(fact_8463_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
tff(fact_8464_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
tff(fact_8465_aset_I2_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)) ) ) ) ) ) ).

% aset(2)
tff(fact_8466_aset_I1_J,axiom,
    ! [D4: int,A3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,A3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,A3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)) ) ) ) ) ) ).

% aset(1)
tff(fact_8467_bset_I2_J,axiom,
    ! [D4: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                | aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))
                | aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)) ) ) ) ) ) ).

% bset(2)
tff(fact_8468_bset_I1_J,axiom,
    ! [D4: int,B3: set_int,P: fun_int_o,Q: fun_int_o] :
      ( ! [X3: int] :
          ( ! [Xa: int] :
              ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb2: int] :
                  ( member_int(Xb2,B3)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
         => ( aa_int_o(P,X3)
           => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
     => ( ! [X3: int] :
            ( ! [Xa: int] :
                ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb2: int] :
                    ( member_int(Xb2,B3)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
           => ( aa_int_o(Q,X3)
             => aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( aa_int_o(P,X4)
                & aa_int_o(Q,X4) )
             => ( aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4))
                & aa_int_o(Q,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)) ) ) ) ) ) ).

% bset(1)
tff(fact_8469_bset_I9_J,axiom,
    ! [D: int,D4: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),Ta)) ) ) ) ).

% bset(9)
tff(fact_8470_bset_I10_J,axiom,
    ! [D: int,D4: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),Ta)) ) ) ) ).

% bset(10)
tff(fact_8471_aset_I9_J,axiom,
    ! [D: int,D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),Ta)) ) ) ) ).

% aset(9)
tff(fact_8472_aset_I10_J,axiom,
    ! [D: int,D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),Ta))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),Ta)) ) ) ) ).

% aset(10)
tff(fact_8473_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_8474_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(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D))) )
       => ( ? [X_13: int] : aa_int_o(P,X_13)
        <=> ? [X: int] :
              ( member_int(X,set_or1266510415728281911st_int(one_one_int,D))
              & aa_int_o(P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_8475_aset_I7_J,axiom,
    ! [D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)) ) ) ) ).

% aset(7)
tff(fact_8476_aset_I5_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_8477_aset_I4_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_8478_aset_I3_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_8479_bset_I7_J,axiom,
    ! [D4: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)) ) ) ) ) ).

% bset(7)
tff(fact_8480_bset_I5_J,axiom,
    ! [D4: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X4),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),Ta) ) ) ) ).

% bset(5)
tff(fact_8481_bset_I4_J,axiom,
    ! [D4: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(Ta,B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X4 != Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_8482_bset_I3_J,axiom,
    ! [D4: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( ( X4 = Ta )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_8483_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_8484_aset_I8_J,axiom,
    ! [D4: int,A3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,A3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)) ) ) ) ).

% aset(8)
tff(fact_8485_aset_I6_J,axiom,
    ! [D4: int,Ta: int,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Ta),one_one_int),A3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,A3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),D4)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_8486_bset_I8_J,axiom,
    ! [D4: int,Ta: int,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( member_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Ta),one_one_int),B3)
       => ! [X4: int] :
            ( ! [Xa2: int] :
                ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
               => ! [Xb3: int] :
                    ( member_int(Xb3,B3)
                   => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),X4)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Ta),aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)) ) ) ) ) ).

% bset(8)
tff(fact_8487_bset_I6_J,axiom,
    ! [D4: int,B3: set_int,Ta: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ! [X4: int] :
          ( ! [Xa2: int] :
              ( member_int(Xa2,set_or1266510415728281911st_int(one_one_int,D4))
             => ! [Xb3: int] :
                  ( member_int(Xb3,B3)
                 => ( X4 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa2) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X4),Ta)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X4),D4)),Ta) ) ) ) ).

% bset(6)
tff(fact_8488_cpmi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,B3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),Z2)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,B3)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X: int] :
                    ( member_int(X,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( member_int(X,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,B3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_8489_cppi,axiom,
    ! [D4: int,P: fun_int_o,P3: fun_int_o,A3: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D4)
     => ( ? [Z2: int] :
          ! [X3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z2),X3)
           => ( aa_int_o(P,X3)
            <=> aa_int_o(P3,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( member_int(Xa,set_or1266510415728281911st_int(one_one_int,D4))
                 => ! [Xb2: int] :
                      ( member_int(Xb2,A3)
                     => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa) ) ) )
             => ( aa_int_o(P,X3)
               => aa_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D4)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa_int_o(P3,X3)
              <=> aa_int_o(P3,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D4))) )
           => ( ? [X_13: int] : aa_int_o(P,X_13)
            <=> ( ? [X: int] :
                    ( member_int(X,set_or1266510415728281911st_int(one_one_int,D4))
                    & aa_int_o(P3,X) )
                | ? [X: int] :
                    ( member_int(X,set_or1266510415728281911st_int(one_one_int,D4))
                    & ? [Xa3: int] :
                        ( member_int(Xa3,A3)
                        & aa_int_o(P,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_8490__092_060open_062res_A_061_Athe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_092_060close_062,axiom,
    res = the_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% \<open>res = the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2)))))\<close>
tff(fact_8491_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya)),Dega)
     => ( ( Mia != Maa )
       => ( the_nat(vEBT_vebt_maxt(Summarya)) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Maa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ) ).

% summaxma
tff(fact_8492_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb)
       => ( vEBT_vebt_maxt(Ta) = some_nat(Xb) ) ) ) ).

% maxt_sound
tff(fact_8493_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
       => aa_nat_o(vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta)),Xb) ) ) ).

% maxt_corr
tff(fact_8494_add__shift,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),Y) = Zb )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,some_nat(Xb)),some_nat(Y)) = some_nat(Zb) ) ) ).

% add_shift
tff(fact_8495_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y) = Zb )
    <=> ( aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(Xb)),some_nat(Y)) = some_nat(Zb) ) ) ).

% mul_shift
tff(fact_8496_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = some_nat(Xb) )
     => aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_8497_add__def,axiom,
    vEBT_VEBT_add = vEBT_V4262088993061758097ft_nat(plus_plus_nat) ).

% add_def
tff(fact_8498_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V4262088993061758097ft_nat(times_times_nat) ).

% mul_def
tff(fact_8499_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_8500_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => ( ( vEBT_vebt_maxt(Ta) = some_nat(Maxi) )
       => ( aa_nat_o(vEBT_vebt_member(Ta),Xb)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_8501_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_8502__092_060open_062vebt__succ_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ASome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_092_060close_062,axiom,
    vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))) ).

% \<open>vebt_succ (Node (Some (mi, ma)) deg treeList summary) x = Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))\<close>
tff(fact_8503__C2_C,axiom,
    vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = $ite(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ).

% "2"
tff(fact_8504__092_060open_062vebt__member_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_A_Ithe_A_ISome_A_I2_A_094_A_Ideg_Adiv_A2_J_J_A_K_092_060_094sub_062o_Avebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_A_L_092_060_094sub_062o_Avebt__mint_A_ItreeList_A_B_Athe_A_Ivebt__succ_Asummary_A_Ihigh_Ax_A_Ideg_Adiv_A2_J_J_J_J_J_J_092_060close_062,axiom,
    aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary)),the_nat(aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))))))) ).

% \<open>vebt_member (Node (Some (mi, ma)) deg treeList summary) (the (Some (2 ^ (deg div 2)) *\<^sub>o vebt_succ summary (high x (deg div 2)) +\<^sub>o vebt_mint (treeList ! the (vebt_succ summary (high x (deg div 2))))))\<close>
tff(fact_8505__C1_C,axiom,
    vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(mi),ma)),deg,treeList,summary),xa) = $ite(
      ( ( vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) != none_nat )
      & vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ),
      aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_succ(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
      $ite(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) = none_nat,none_nat,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_add,aa_opt1882420858203594238on_nat(aa_opt7711074727033721499on_nat(vEBT_VEBT_mul,some_nat(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),vEBT_vebt_mint(nth_VEBT_VEBT(treeList,the_nat(vEBT_vebt_succ(summary,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))))) ) ).

% "1"
tff(fact_8506_i1,axiom,
    ( ( vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) = none_nat )
    | ~ vEBT_VEBT_less(some_nat(aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),vEBT_vebt_maxt(nth_VEBT_VEBT(treeList,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% i1
tff(fact_8507_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_8508_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),aa_num_int(numeral_numeral_int,L))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_8509_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_8510_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y)
    <=> vEBT_VEBT_less(some_nat(Xb),some_nat(Y)) ) ).

% less_shift
tff(fact_8511_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc8114805097830739791at_num] :
      ~ ! [F2: fun_nat_fun_num_num,A4: nat,B4: nat,Acc2: num] : Xb != produc460779904311030791at_num(F2,produc1195630363706982562at_num(A4,product_Pair_nat_num(B4,Acc2))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_8512_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [Xb: produc9217583074051984497at_nat] :
      ~ ! [F2: fun_nat_fun_nat_nat,A4: nat,B4: nat,Acc2: nat] : Xb != produc2818902965508115497at_nat(F2,produc487386426758144856at_nat(A4,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B4),Acc2))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_8513_xor__num_Ocases,axiom,
    ! [Xb: product_prod_num_num] :
      ( ( Xb != product_Pair_num_num(one,one) )
     => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit0,N2))
       => ( ! [N2: num] : Xb != product_Pair_num_num(one,aa_num_num(bit1,N2))
         => ( ! [M3: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M3),one)
           => ( ! [M3: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))
             => ( ! [M3: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))
               => ( ! [M3: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M3),one)
                 => ( ! [M3: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))
                   => ~ ! [M3: num,N2: num] : Xb != product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_8514_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ( vEBT_vebt_mint(Ta) = none_nat ) ) ).

% minNullmin
tff(fact_8515_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none_nat )
     => aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta) ) ).

% minminNull
tff(fact_8516_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] : ~ aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(V),aa_nat_nat(suc,zero_zero_nat),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_8517_succ__less__length__list,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,TreeLista: list_VEBT_VEBT,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
         => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & 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(
                        sc2: option_nat,
                        sc2:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc2 = 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)))))),sc2)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc2))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_8518_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_8519_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_VEBT_VEBT_o(vEBT_VEBT_minNull,Ta)
     => ~ aa_nat_o(vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_8520_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect_nat(vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_8521_succ__greatereq__min,axiom,
    ! [Dega: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mia),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
              $let(
                h: nat,
                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                $ite(
                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),
                  $let(
                    maxlow: option_nat,
                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeLista,h)),
                    $ite(
                      ( ( maxlow != none_nat )
                      & 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(
                        sc2: option_nat,
                        sc2:= vEBT_vebt_succ(Summarya,h),
                        $ite(sc2 = 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)))))),sc2)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc2))))) ) ) ),
                  none_nat ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_8522_nat__less__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_less_as_int
tff(fact_8523_less__set__def,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat] :
      ( ord_le2529575680413868914d_enat(A3,B3)
    <=> ord_le674264225683014473enat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B3)) ) ).

% less_set_def
tff(fact_8524_less__set__def,axiom,
    ! [A3: set_complex,B3: set_complex] :
      ( ord_less_set_complex(A3,B3)
    <=> ord_le4409593196786536943plex_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3)) ) ).

% less_set_def
tff(fact_8525_less__set__def,axiom,
    ! [A3: set_real,B3: set_real] :
      ( ord_less_set_real(A3,B3)
    <=> ord_less_fun_real_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3)) ) ).

% less_set_def
tff(fact_8526_less__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ord_less_set_nat(A3,B3)
    <=> ord_less_fun_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3)) ) ).

% less_set_def
tff(fact_8527_less__set__def,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ord_less_set_int(A3,B3)
    <=> ord_less_fun_int_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3)) ) ).

% less_set_def
tff(fact_8528_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ord_less_set_complex(collect_complex(aTP_Lamm_af(A)),collect_complex(aTP_Lamm_af(B2)))
    <=> ( aa_complex_o(dvd_dvd_complex(A),B2)
        & ~ aa_complex_o(dvd_dvd_complex(B2),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8529_strict__subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ord_less_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_ag,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_ag,B2)))
    <=> ( aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2)
        & ~ aa_real_o(aa_real_fun_real_o(dvd_dvd_real,B2),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8530_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ord_less_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah,A)),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah,B2)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2)
        & ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B2),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8531_strict__subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ord_less_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ai,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ai,B2)))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2)
        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B2),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8532_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ord_le1307284697595431911nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aj,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aj,B2)))
    <=> ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2)
        & ~ aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,B2),A) ) ) ).

% strict_subset_divisors_dvd
tff(fact_8533_set__diff__eq,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat] : minus_925952699566721837d_enat(A3,B3) = collec4429806609662206161d_enat(aa_set3572156908926473983enat_o(aTP_Lamm_ak(A3),B3)) ).

% set_diff_eq
tff(fact_8534_set__diff__eq,axiom,
    ! [A3: set_complex,B3: set_complex] : minus_811609699411566653omplex(A3,B3) = collect_complex(aa_set1742385355640705825plex_o(aTP_Lamm_al(A3),B3)) ).

% set_diff_eq
tff(fact_8535_set__diff__eq,axiom,
    ! [A3: set_real,B3: set_real] : minus_minus_set_real(A3,B3) = collect_real(aa_set2323979645938365349real_o(aTP_Lamm_am(A3),B3)) ).

% set_diff_eq
tff(fact_8536_set__diff__eq,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A3,B3) = collect_list_nat(aa_set6675653781276780861_nat_o(aTP_Lamm_an(A3),B3)) ).

% set_diff_eq
tff(fact_8537_set__diff__eq,axiom,
    ! [A3: set_int,B3: set_int] : minus_minus_set_int(A3,B3) = collect_int(aa_set_int_fun_int_o(aTP_Lamm_ao(A3),B3)) ).

% set_diff_eq
tff(fact_8538_set__diff__eq,axiom,
    ! [A3: set_nat,B3: set_nat] : minus_minus_set_nat(A3,B3) = collect_nat(aa_set_nat_fun_nat_o(aTP_Lamm_ap(A3),B3)) ).

% set_diff_eq
tff(fact_8539_minus__set__def,axiom,
    ! [A3: set_Extended_enat,B3: set_Extended_enat] : minus_925952699566721837d_enat(A3,B3) = collec4429806609662206161d_enat(minus_6189366049226783950enat_o(aTP_Lamm_a(A3),aTP_Lamm_a(B3))) ).

% minus_set_def
tff(fact_8540_minus__set__def,axiom,
    ! [A3: set_complex,B3: set_complex] : minus_811609699411566653omplex(A3,B3) = collect_complex(minus_8169481405280535082plex_o(aTP_Lamm_aa(A3),aTP_Lamm_aa(B3))) ).

% minus_set_def
tff(fact_8541_minus__set__def,axiom,
    ! [A3: set_real,B3: set_real] : minus_minus_set_real(A3,B3) = collect_real(minus_404528632214381484real_o(aTP_Lamm_ab(A3),aTP_Lamm_ab(B3))) ).

% minus_set_def
tff(fact_8542_minus__set__def,axiom,
    ! [A3: set_list_nat,B3: set_list_nat] : minus_7954133019191499631st_nat(A3,B3) = collect_list_nat(minus_5761417318614613624_nat_o(aTP_Lamm_ac(A3),aTP_Lamm_ac(B3))) ).

% minus_set_def
tff(fact_8543_minus__set__def,axiom,
    ! [A3: set_int,B3: set_int] : minus_minus_set_int(A3,B3) = collect_int(minus_7151557278095657260_int_o(aTP_Lamm_ae(A3),aTP_Lamm_ae(B3))) ).

% minus_set_def
tff(fact_8544_minus__set__def,axiom,
    ! [A3: set_nat,B3: set_nat] : minus_minus_set_nat(A3,B3) = collect_nat(minus_8972599623528345224_nat_o(aTP_Lamm_ad(A3),aTP_Lamm_ad(B3))) ).

% minus_set_def
tff(fact_8545_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_8546_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_8547_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(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2) ) ).

% numeral_code(2)
tff(fact_8548_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_8549_numeral__code_I2_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2) ) ).

% numeral_code(2)
tff(fact_8550_lambda__zero,axiom,
    aTP_Lamm_aq = aa_rea7340605660300375871l_real(times_times_real,zero_zero_real) ).

% lambda_zero
tff(fact_8551_lambda__zero,axiom,
    aTP_Lamm_ar = aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat) ).

% lambda_zero
tff(fact_8552_lambda__zero,axiom,
    aTP_Lamm_as = aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat) ).

% lambda_zero
tff(fact_8553_lambda__zero,axiom,
    aTP_Lamm_at = aa_int_fun_int_int(times_times_int,zero_zero_int) ).

% lambda_zero
tff(fact_8554_mult__commute__abs,axiom,
    ! [C: real] : aa_rea7340605660300375871l_real(aTP_Lamm_au,C) = aa_rea7340605660300375871l_real(times_times_real,C) ).

% mult_commute_abs
tff(fact_8555_mult__commute__abs,axiom,
    ! [C: rat] : aTP_Lamm_av(C) = aa_rat_fun_rat_rat(times_times_rat,C) ).

% mult_commute_abs
tff(fact_8556_mult__commute__abs,axiom,
    ! [C: nat] : aa_nat_fun_nat_nat(aTP_Lamm_aw,C) = aa_nat_fun_nat_nat(times_times_nat,C) ).

% mult_commute_abs
tff(fact_8557_mult__commute__abs,axiom,
    ! [C: int] : aTP_Lamm_ax(C) = aa_int_fun_int_int(times_times_int,C) ).

% mult_commute_abs
tff(fact_8558_lambda__one,axiom,
    aTP_Lamm_ay = times_times_complex(one_one_complex) ).

% lambda_one
tff(fact_8559_lambda__one,axiom,
    aTP_Lamm_az = aa_rea7340605660300375871l_real(times_times_real,one_one_real) ).

% lambda_one
tff(fact_8560_lambda__one,axiom,
    aTP_Lamm_ba = aa_rat_fun_rat_rat(times_times_rat,one_one_rat) ).

% lambda_one
tff(fact_8561_lambda__one,axiom,
    aTP_Lamm_bb = aa_nat_fun_nat_nat(times_times_nat,one_one_nat) ).

% lambda_one
tff(fact_8562_lambda__one,axiom,
    aTP_Lamm_bc = aa_int_fun_int_int(times_times_int,one_one_int) ).

% lambda_one
tff(fact_8563_subset__divisors__dvd,axiom,
    ! [A: complex,B2: complex] :
      ( ord_le211207098394363844omplex(collect_complex(aTP_Lamm_af(A)),collect_complex(aTP_Lamm_af(B2)))
    <=> aa_complex_o(dvd_dvd_complex(A),B2) ) ).

% subset_divisors_dvd
tff(fact_8564_subset__divisors__dvd,axiom,
    ! [A: real,B2: real] :
      ( ord_less_eq_set_real(collect_real(aa_real_fun_real_o(aTP_Lamm_ag,A)),collect_real(aa_real_fun_real_o(aTP_Lamm_ag,B2)))
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,A),B2) ) ).

% subset_divisors_dvd
tff(fact_8565_subset__divisors__dvd,axiom,
    ! [A: int,B2: int] :
      ( ord_less_eq_set_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ai,A)),collect_int(aa_int_fun_int_o(aTP_Lamm_ai,B2)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B2) ) ).

% subset_divisors_dvd
tff(fact_8566_subset__divisors__dvd,axiom,
    ! [A: code_integer,B2: code_integer] :
      ( ord_le7084787975880047091nteger(collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aj,A)),collect_Code_integer(aa_Cod326357343816142457eger_o(aTP_Lamm_aj,B2)))
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,A),B2) ) ).

% subset_divisors_dvd
tff(fact_8567_subset__divisors__dvd,axiom,
    ! [A: nat,B2: nat] :
      ( ord_less_eq_set_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah,A)),collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah,B2)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B2) ) ).

% subset_divisors_dvd
tff(fact_8568_nat__leq__as__int,axiom,
    ! [X4: nat,Xa: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Xa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa)) ) ).

% nat_leq_as_int
tff(fact_8569_power__numeral__even,axiom,
    ! [Zb: complex,W: num] :
      aa_nat_complex(power_power_complex(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(w),w) ) ).

% power_numeral_even
tff(fact_8570_power__numeral__even,axiom,
    ! [Zb: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),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,Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_real_real(aa_rea7340605660300375871l_real(times_times_real,w),w) ) ).

% power_numeral_even
tff(fact_8571_power__numeral__even,axiom,
    ! [Zb: rat,W: num] :
      aa_nat_rat(power_power_rat(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,w),w) ) ).

% power_numeral_even
tff(fact_8572_power__numeral__even,axiom,
    ! [Zb: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Zb),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,Zb),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_8573_power__numeral__even,axiom,
    ! [Zb: int,W: num] :
      aa_nat_int(power_power_int(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,w),w) ) ).

% power_numeral_even
tff(fact_8574_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_8575_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_8576_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(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,m2),m2)),one_one_rat) ) ).

% numeral_code(3)
tff(fact_8577_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_8578_numeral__code_I3_J,axiom,
    ! [Nb: num] :
      aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)) = $let(
        m2: int,
        m2:= aa_num_int(numeral_numeral_int,Nb),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,m2),m2)),one_one_int) ) ).

% numeral_code(3)
tff(fact_8579_power__numeral__odd,axiom,
    ! [Zb: complex,W: num] :
      aa_nat_complex(power_power_complex(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: complex,
        w:= aa_nat_complex(power_power_complex(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_complex_complex(times_times_complex(aa_complex_complex(times_times_complex(Zb),w)),w) ) ).

% power_numeral_odd
tff(fact_8580_power__numeral__odd,axiom,
    ! [Zb: real,W: num] :
      aa_nat_real(aa_real_fun_nat_real(power_power_real,Zb),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,Zb),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,Zb),w)),w) ) ).

% power_numeral_odd
tff(fact_8581_power__numeral__odd,axiom,
    ! [Zb: rat,W: num] :
      aa_nat_rat(power_power_rat(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: rat,
        w:= aa_nat_rat(power_power_rat(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Zb),w)),w) ) ).

% power_numeral_odd
tff(fact_8582_power__numeral__odd,axiom,
    ! [Zb: nat,W: num] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Zb),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,Zb),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,Zb),w)),w) ) ).

% power_numeral_odd
tff(fact_8583_power__numeral__odd,axiom,
    ! [Zb: int,W: num] :
      aa_nat_int(power_power_int(Zb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,W))) = $let(
        w: int,
        w:= aa_nat_int(power_power_int(Zb),aa_num_nat(numeral_numeral_nat,W)),
        aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,Zb),w)),w) ) ).

% power_numeral_odd
tff(fact_8584_nat__plus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_plus_as_int
tff(fact_8585_nat__times__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_times_as_int
tff(fact_8586_nat__minus__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_minus_as_int
tff(fact_8587_nat__div__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_div_as_int
tff(fact_8588_nat__mod__as__int,axiom,
    ! [X4: nat,Xa: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X4),Xa) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(semiri1314217659103216013at_int,X4)),aa_nat_int(semiri1314217659103216013at_int,Xa))) ).

% nat_mod_as_int
tff(fact_8589_set__conv__nth,axiom,
    ! [Xs: list_complex] : set_complex2(Xs) = collect_complex(aTP_Lamm_bd(Xs)) ).

% set_conv_nth
tff(fact_8590_set__conv__nth,axiom,
    ! [Xs: list_real] : set_real2(Xs) = collect_real(aTP_Lamm_be(Xs)) ).

% set_conv_nth
tff(fact_8591_set__conv__nth,axiom,
    ! [Xs: list_list_nat] : set_list_nat2(Xs) = collect_list_nat(aTP_Lamm_bf(Xs)) ).

% set_conv_nth
tff(fact_8592_set__conv__nth,axiom,
    ! [Xs: list_VEBT_VEBT] : set_VEBT_VEBT2(Xs) = collect_VEBT_VEBT(aTP_Lamm_bg(Xs)) ).

% set_conv_nth
tff(fact_8593_set__conv__nth,axiom,
    ! [Xs: list_o] : set_o2(Xs) = collect_o(aTP_Lamm_bh(Xs)) ).

% set_conv_nth
tff(fact_8594_set__conv__nth,axiom,
    ! [Xs: list_nat] : set_nat2(Xs) = collect_nat(aTP_Lamm_bi(Xs)) ).

% set_conv_nth
tff(fact_8595_set__conv__nth,axiom,
    ! [Xs: list_int] : set_int2(Xs) = collect_int(aTP_Lamm_bj(Xs)) ).

% set_conv_nth
tff(fact_8596_diff__nat__eq__if,axiom,
    ! [Zb: int,Z3: int] :
      aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_int_nat(nat2,Zb)),aa_int_nat(nat2,Z3)) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,Z3),zero_zero_int),
        aa_int_nat(nat2,Zb),
        $let(
          d: int,
          d:= aa_int_int(aa_int_fun_int_int(minus_minus_int,Zb),Z3),
          $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_8597_set__decode__def,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,Xb) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bk,Xb)) ).

% set_decode_def
tff(fact_8598_signed__take__bit__code,axiom,
    ! [Nb: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(Nb,A) = $let(
        l: code_integer,
        l:= bit_se1745604003318907178nteger(aa_nat_nat(suc,Nb),A),
        $ite(aa_nat_o(bit_se9216721137139052372nteger(l),Nb),plus_p5714425477246183910nteger(l,bit_se7788150548672797655nteger(aa_nat_nat(suc,Nb),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))),l) ) ).

% signed_take_bit_code
tff(fact_8599_signed__take__bit__code,axiom,
    ! [Nb: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(Nb),A) = $let(
        l: int,
        l:= aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,Nb)),A),
        $ite(aa_nat_o(bit_se1146084159140164899it_int(l),Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,l),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,Nb)),aa_int_int(uminus_uminus_int,one_one_int))),l) ) ).

% signed_take_bit_code
tff(fact_8600_pochhammer__code,axiom,
    ! [A: complex,Nb: nat] :
      aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = $ite(Nb = zero_zero_nat,one_one_complex,aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_bl(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_complex)) ).

% pochhammer_code
tff(fact_8601_pochhammer__code,axiom,
    ! [A: real,Nb: nat] :
      aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = $ite(Nb = zero_zero_nat,one_one_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_bm(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_real)) ).

% pochhammer_code
tff(fact_8602_pochhammer__code,axiom,
    ! [A: rat,Nb: nat] :
      aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = $ite(Nb = zero_zero_nat,one_one_rat,aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_bn(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_rat)) ).

% pochhammer_code
tff(fact_8603_pochhammer__code,axiom,
    ! [A: int,Nb: nat] :
      aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = $ite(Nb = zero_zero_nat,one_one_int,aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_bo(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_int)) ).

% pochhammer_code
tff(fact_8604_pochhammer__code,axiom,
    ! [A: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = $ite(Nb = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_bp(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),one_one_nat)) ).

% pochhammer_code
tff(fact_8605_gbinomial__code,axiom,
    ! [A: complex,K: nat] :
      aa_nat_complex(gbinomial_complex(A),K) = $ite(K = zero_zero_nat,one_one_complex,aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_bq(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_complex)),aa_nat_complex(semiri5044797733671781792omplex,K))) ).

% gbinomial_code
tff(fact_8606_gbinomial__code,axiom,
    ! [A: rat,K: nat] :
      aa_nat_rat(gbinomial_rat(A),K) = $ite(K = zero_zero_nat,one_one_rat,aa_rat_rat(divide_divide_rat(aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_br(A),zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_rat)),aa_nat_rat(semiri773545260158071498ct_rat,K))) ).

% gbinomial_code
tff(fact_8607_gbinomial__code,axiom,
    ! [A: real,K: nat] :
      aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),K) = $ite(K = zero_zero_nat,one_one_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_bs(A),zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),one_one_nat)),one_one_real)),aa_nat_real(semiri2265585572941072030t_real,K))) ).

% gbinomial_code
tff(fact_8608_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
              $false,
              $ite(
                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Maa),Xb),
                $false,
                $let(
                  h: nat,
                  h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                  $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeLista,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_8609_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xb) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),
        some_nat(Mia),
        $let(
          l: nat,
          l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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,Xb),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 )
                  & 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(
                    sc2: option_nat,
                    sc2:= vEBT_vebt_succ(Summarya,h),
                    $ite(sc2 = 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)))))),sc2)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeLista,the_nat(sc2))))) ) ) ),
              none_nat ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_8610_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_real(ring_1_of_int_real,K) = $ite(
        K = zero_zero_int,
        zero_zero_real,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_real_real(uminus_uminus_real,aa_int_real(ring_1_of_int_real,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: real,
            l:= aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,l),one_one_real)) ) ) ) ).

% of_int_code_if
tff(fact_8611_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_int(ring_1_of_int_int,K) = $ite(
        K = zero_zero_int,
        zero_zero_int,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_int_int(uminus_uminus_int,aa_int_int(ring_1_of_int_int,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: int,
            l:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(ring_1_of_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_int_int(aa_int_fun_int_int(plus_plus_int,l),one_one_int)) ) ) ) ).

% of_int_code_if
tff(fact_8612_of__int__code__if,axiom,
    ! [K: int] :
      aa_int_complex(ring_17405671764205052669omplex,K) = $ite(
        K = zero_zero_int,
        zero_zero_complex,
        $ite(
          aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
          aa_complex_complex(uminus1482373934393186551omplex,aa_int_complex(ring_17405671764205052669omplex,aa_int_int(uminus_uminus_int,K))),
          $let(
            l: complex,
            l:= aa_complex_complex(times_times_complex(numera6690914467698888265omplex(aa_num_num(bit0,one))),aa_int_complex(ring_17405671764205052669omplex,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_complex_complex(plus_plus_complex(l),one_one_complex)) ) ) ) ).

% of_int_code_if
tff(fact_8613_of__int__code__if,axiom,
    ! [K: int] :
      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,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))),ring_18347121197199848620nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% of_int_code_if
tff(fact_8614_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(aa_rat_fun_rat_rat(times_times_rat,aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,l),one_one_rat)) ) ) ) ).

% of_int_code_if
tff(fact_8615_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list_VEBT_VEBT,Vc: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,V),TreeLista,Vc)),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $let(
            pos: nat,
            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_8616_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => topolo6980174941875973593q_real(aa_real_fun_nat_real(aTP_Lamm_bt,Xb)) ) ).

% monoseq_arctan_series
tff(fact_8617_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeLista: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Node(Uy,aa_nat_nat(suc,V),TreeLista,S)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_8618_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_min_max
tff(fact_8619_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ aa_nat_o(vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb)),Xb) ).

% buildup_nothing_in_leaf
tff(fact_8620_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( aa_nat_o(vEBT_V5719532721284313246member(Ta),Xb)
        | aa_nat_o(vEBT_VEBT_membermima(Ta),Xb) ) ) ).

% both_member_options_def
tff(fact_8621_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Tree),Nb)
     => ( aa_nat_o(vEBT_vebt_member(Tree),Xb)
       => ( aa_nat_o(vEBT_V5719532721284313246member(Tree),Xb)
          | aa_nat_o(vEBT_VEBT_membermima(Tree),Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_8622_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo6980174941875973593q_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% monoseq_realpow
tff(fact_8623_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_8624_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_8625_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S),Xb) = vEBT_Node(Info,aa_nat_nat(suc,zero_zero_nat),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_8626_pochhammer__times__pochhammer__half,axiom,
    ! [Zb: complex,Nb: nat] : aa_complex_complex(times_times_complex(aa_nat_complex(comm_s2602460028002588243omplex(Zb),aa_nat_nat(suc,Nb))),aa_nat_complex(comm_s2602460028002588243omplex(aa_complex_complex(plus_plus_complex(Zb),aa_complex_complex(divide1717551699836669952omplex(one_one_complex),numera6690914467698888265omplex(aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups6464643781859351333omplex(aTP_Lamm_bu(Zb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8627_pochhammer__times__pochhammer__half,axiom,
    ! [Zb: real,Nb: nat] : aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,Zb),aa_nat_nat(suc,Nb))),aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Zb),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,Nb))) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_bv,Zb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8628_pochhammer__times__pochhammer__half,axiom,
    ! [Zb: rat,Nb: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(comm_s4028243227959126397er_rat(Zb),aa_nat_nat(suc,Nb))),aa_nat_rat(comm_s4028243227959126397er_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Zb),aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one))))),aa_nat_nat(suc,Nb))) = groups73079841787564623at_rat(aTP_Lamm_bw(Zb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),one_one_nat))) ).

% pochhammer_times_pochhammer_half
tff(fact_8629_Leaf__0__not,axiom,
    ! [A: $o,B2: $o] : ~ aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B2))),zero_zero_nat) ).

% Leaf_0_not
tff(fact_8630_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_8631_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_8632_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),one_one_nat)
    <=> ? [A5: $o,B6: $o] : Ta = vEBT_Leaf((A5),(B6)) ) ).

% deg1Leaf
tff(fact_8633_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_complex,Mb: nat,Nb: nat] :
      groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_complex,aa_complex_complex(times_times_complex(groups6464643781859351333omplex(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_complex(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8634_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] :
      groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8635_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] :
      groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8636_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] :
      groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8637_prod_Ocl__ivl__Suc,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] :
      groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,Nb)),Mb),one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb)))) ).

% prod.cl_ivl_Suc
tff(fact_8638_mod__prod__eq,axiom,
    ! [F: fun_nat_nat,A: nat,A3: set_nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_bx(F),A),A3)),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,groups708209901874060359at_nat(F,A3)),A) ).

% mod_prod_eq
tff(fact_8639_mod__prod__eq,axiom,
    ! [F: fun_nat_int,A: int,A3: set_nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(aa_int_fun_nat_int(aTP_Lamm_by(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups705719431365010083at_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_8640_mod__prod__eq,axiom,
    ! [F: fun_int_int,A: int,A3: set_int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(aa_int_fun_int_int(aTP_Lamm_bz(F),A),A3)),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,groups1705073143266064639nt_int(F,A3)),A) ).

% mod_prod_eq
tff(fact_8641_vebt__insert_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),X3)
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),X3)
         => ( ! [V3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary),X3)
           => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_8642_vebt__insert_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A),(B2)),Xb) = $ite(
        Xb = zero_zero_nat,
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one_nat,vEBT_Leaf((A),$true),vEBT_Leaf((A),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_8643_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups708209901874060359at_nat(aTP_Lamm_ca(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8644_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb))) = groups705719431365010083at_int(aTP_Lamm_cb(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_8645_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_nat,Mb: nat,K: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cc(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8646_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: fun_nat_int,Mb: nat,K: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),K))) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_cd(G),K),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_8647_vebt__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),X3)
       => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz),X3)
         => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),X3)
           => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_8648_vebt__succ_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B4: $o] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(B4)),zero_zero_nat)
     => ( ! [Uv: $o,Uw: $o,N2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N2))
       => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,Va3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz),Va3)
         => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Ve2)
           => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Vi2)
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),X3) ) ) ) ) ) ).

% vebt_succ.cases
tff(fact_8649_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Uw)
     => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Uz)
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),X3)
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2),X3)
           => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_8650_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Leaf((A4),(B4)),X3)
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Ux)
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X3: nat] : Xb != produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_8651_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B2: $o] : aa_nat_o(vEBT_invar_vebt(vEBT_Leaf((A),(B2))),aa_nat_nat(suc,zero_zero_nat)) ).

% invar_vebt.intros(1)
tff(fact_8652_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xb: nat] :
      ( aa_nat_o(vEBT_vebt_member(vEBT_Leaf((A),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_8653_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_8654_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa_nat_nat(suc,Nb)) = none_nat ).

% vebt_succ.simps(2)
tff(fact_8655_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o,Xb: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(vEBT_Leaf((A),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero_nat,
          (A),
          $ite(Xb = one_one_nat,(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_8656_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_nat,Nb: nat,Mb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ce(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_8657_prod_OatLeastAtMost__rev,axiom,
    ! [G: fun_nat_int,Nb: nat,Mb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_cf(G),Nb),Mb),aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb)) ).

% prod.atLeastAtMost_rev
tff(fact_8658_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux2: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] : vEBT_vebt_succ(vEBT_Node(none_P5556105721700978146at_nat,Ux2,Uy,Uz2),Va2) = none_nat ).

% vebt_succ.simps(3)
tff(fact_8659_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_real,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8660_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_rat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8661_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8662_prod_OatLeast0__atMost__Suc,axiom,
    ! [G: fun_nat_int,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) ).

% prod.atLeast0_atMost_Suc
tff(fact_8663_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8664_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8665_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8666_prod_OatLeast__Suc__atMost,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb))) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_8667_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,aa_nat_nat(suc,Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8668_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,aa_nat_nat(suc,Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8669_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,aa_nat_nat(suc,Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8670_prod_Onat__ivl__Suc_H,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,aa_nat_nat(suc,Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_8671_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xb)),aa_nat_nat(suc,aa_nat_nat(suc,V)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_8672_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_real(G,aa_nat_nat(suc,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(G,Mb)),groups129246275422532515t_real(aTP_Lamm_cg(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8673_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_rat(G,aa_nat_nat(suc,Nb))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(G,Mb)),groups73079841787564623at_rat(aTP_Lamm_ch(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8674_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_nat(G,aa_nat_nat(suc,Nb))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(G,Mb)),groups708209901874060359at_nat(aTP_Lamm_ca(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8675_prod_OSuc__reindex__ivl,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),aa_nat_int(G,aa_nat_nat(suc,Nb))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(G,Mb)),groups705719431365010083at_int(aTP_Lamm_cb(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_8676_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_complex(semiri5044797733671781792omplex,Nb) = aa_nat_complex(semiri8010041392384452111omplex,groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8677_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_rat(semiri773545260158071498ct_rat,Nb) = aa_nat_rat(semiri681578069525770553at_rat,groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8678_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_int(semiri1406184849735516958ct_int,Nb) = aa_nat_int(semiri1314217659103216013at_int,groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8679_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1408675320244567234ct_nat,Nb) = aa_nat_nat(semiri1316708129612266289at_nat,groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8680_fact__prod,axiom,
    ! [Nb: nat] : aa_nat_real(semiri2265585572941072030t_real,Nb) = aa_nat_real(semiri5074537144036343181t_real,groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb))) ).

% fact_prod
tff(fact_8681_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_complex,A: nat,B2: nat] : groups6464643781859351333omplex(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_complex_complex(set_fo1517530859248394432omplex(aTP_Lamm_ci(F),A,B2),one_one_complex) ).

% prod_atLeastAtMost_code
tff(fact_8682_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_real,A: nat,B2: nat] : groups129246275422532515t_real(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_real_real(aa_nat_fun_real_real(set_fo3111899725591712190t_real(aTP_Lamm_cj(F),A),B2),one_one_real) ).

% prod_atLeastAtMost_code
tff(fact_8683_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_rat,A: nat,B2: nat] : groups73079841787564623at_rat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_rat_rat(set_fo1949268297981939178at_rat(aTP_Lamm_ck(F),A,B2),one_one_rat) ).

% prod_atLeastAtMost_code
tff(fact_8684_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_nat,A: nat,B2: nat] : groups708209901874060359at_nat(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_nat_nat(aa_nat_fun_nat_nat(set_fo2584398358068434914at_nat(aTP_Lamm_cl(F),A),B2),one_one_nat) ).

% prod_atLeastAtMost_code
tff(fact_8685_prod__atLeastAtMost__code,axiom,
    ! [F: fun_nat_int,A: nat,B2: nat] : groups705719431365010083at_int(F,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = aa_int_int(set_fo2581907887559384638at_int(aTP_Lamm_cm(F),A,B2),one_one_int) ).

% prod_atLeastAtMost_code
tff(fact_8686_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_real,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8687_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_rat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8688_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_nat,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8689_prod_Oub__add__nat,axiom,
    ! [Mb: nat,Nb: nat,G: fun_nat_int,P2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))
     => ( groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2))) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb))),groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),P2)))) ) ) ).

% prod.ub_add_nat
tff(fact_8690_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu2: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu2),(B2)),zero_zero_nat) = $ite((B2),some_nat(one_one_nat),none_nat) ).

% vebt_succ.simps(1)
tff(fact_8691_fact__eq__fact__times,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(semiri1408675320244567234ct_nat,Mb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Nb)),groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Nb)),Mb))) ) ) ).

% fact_eq_fact_times
tff(fact_8692_pochhammer__Suc__prod,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = groups6464643781859351333omplex(aTP_Lamm_cn(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8693_pochhammer__Suc__prod,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_co,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8694_pochhammer__Suc__prod,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aTP_Lamm_cp(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8695_pochhammer__Suc__prod,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cq,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8696_pochhammer__Suc__prod,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aTP_Lamm_cr(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod
tff(fact_8697_pochhammer__prod__rev,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),Nb) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_cs(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8698_pochhammer__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),Nb) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ct(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8699_pochhammer__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),Nb) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_cu(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8700_pochhammer__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cv(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8701_pochhammer__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_cw(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),Nb)) ).

% pochhammer_prod_rev
tff(fact_8702_fact__div__fact,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,Mb)),aa_nat_nat(semiri1408675320244567234ct_nat,Nb)) = groups708209901874060359at_nat(aTP_Lamm_bb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)),Mb)) ) ) ).

% fact_div_fact
tff(fact_8703_prod_Oin__pairs,axiom,
    ! [G: fun_nat_real,Mb: nat,Nb: nat] : groups129246275422532515t_real(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups129246275422532515t_real(aTP_Lamm_cx(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8704_prod_Oin__pairs,axiom,
    ! [G: fun_nat_rat,Mb: nat,Nb: nat] : groups73079841787564623at_rat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups73079841787564623at_rat(aTP_Lamm_cy(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8705_prod_Oin__pairs,axiom,
    ! [G: fun_nat_nat,Mb: nat,Nb: nat] : groups708209901874060359at_nat(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups708209901874060359at_nat(aTP_Lamm_cz(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8706_prod_Oin__pairs,axiom,
    ! [G: fun_nat_int,Mb: nat,Nb: nat] : groups705719431365010083at_int(G,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)))) = groups705719431365010083at_int(aTP_Lamm_da(G),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ).

% prod.in_pairs
tff(fact_8707_pochhammer__Suc__prod__rev,axiom,
    ! [A: complex,Nb: nat] : aa_nat_complex(comm_s2602460028002588243omplex(A),aa_nat_nat(suc,Nb)) = groups6464643781859351333omplex(aa_nat6454746962644518025omplex(aTP_Lamm_cs(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8708_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,Nb: nat] : aa_nat_real(aa_real_fun_nat_real(comm_s7457072308508201937r_real,A),aa_nat_nat(suc,Nb)) = groups129246275422532515t_real(aa_nat_fun_nat_real(aTP_Lamm_ct(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8709_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,Nb: nat] : aa_nat_rat(comm_s4028243227959126397er_rat(A),aa_nat_nat(suc,Nb)) = groups73079841787564623at_rat(aa_nat_fun_nat_rat(aTP_Lamm_cu(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8710_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(comm_s4663373288045622133er_nat,A),aa_nat_nat(suc,Nb)) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_cv(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8711_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,Nb: nat] : aa_nat_int(comm_s4660882817536571857er_int(A),aa_nat_nat(suc,Nb)) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_cw(A),Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb)) ).

% pochhammer_Suc_prod_rev
tff(fact_8712_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => (Y) )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_8713_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( ! [Uu: $o,Uv: $o] : Xb != vEBT_Leaf((Uu),(Uv))
       => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2)
                   => $let(
                        pos: nat,
                        pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_8714_gbinomial__Suc,axiom,
    ! [A: complex,K: nat] : aa_nat_complex(gbinomial_complex(A),aa_nat_nat(suc,K)) = aa_complex_complex(divide1717551699836669952omplex(groups6464643781859351333omplex(aTP_Lamm_db(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_complex(semiri5044797733671781792omplex,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8715_gbinomial__Suc,axiom,
    ! [A: rat,K: nat] : aa_nat_rat(gbinomial_rat(A),aa_nat_nat(suc,K)) = aa_rat_rat(divide_divide_rat(groups73079841787564623at_rat(aTP_Lamm_dc(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_rat(semiri773545260158071498ct_rat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8716_gbinomial__Suc,axiom,
    ! [A: real,K: nat] : aa_nat_real(aa_real_fun_nat_real(gbinomial_real,A),aa_nat_nat(suc,K)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,groups129246275422532515t_real(aa_real_fun_nat_real(aTP_Lamm_dd,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8717_gbinomial__Suc,axiom,
    ! [A: nat,K: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gbinomial_nat,A),aa_nat_nat(suc,K)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_de,A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_nat(semiri1408675320244567234ct_nat,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8718_gbinomial__Suc,axiom,
    ! [A: int,K: nat] : aa_nat_int(gbinomial_int(A),aa_nat_nat(suc,K)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,groups705719431365010083at_int(aTP_Lamm_df(A),aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),K))),aa_nat_int(semiri1406184849735516958ct_int,aa_nat_nat(suc,K))) ).

% gbinomial_Suc
tff(fact_8719_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => (Y) )
         => ( ( ? [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz)
             => (Y) )
           => ( ( ? [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_8720_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
         => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz)
           => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] : Xb != vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2)
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_8721_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( aa_nat_o(vEBT_invar_vebt(A1),A22)
    <=> ( ( ? [A5: $o,B6: $o] : A1 = vEBT_Leaf((A5),(B6))
          & ( A22 = aa_nat_nat(suc,zero_zero_nat) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList2: list_VEBT_VEBT,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),N4)
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),N4) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),N4) )
            & ! [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))),N4))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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))),N4))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N4) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N4)) )
                       => ( 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,N4: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
               => aa_nat_o(vEBT_invar_vebt(X),N4) )
            & aa_nat_o(vEBT_invar_vebt(Summary2),aa_nat_nat(suc,N4))
            & ( size_s6755466524823107622T_VEBT(TreeList2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(suc,N4)) )
            & ( A22 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N4),aa_nat_nat(suc,N4)) )
            & ! [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))),aa_nat_nat(suc,N4)))
               => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),X_13)
                <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList2))
                 => ~ ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi2),Ma2)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [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))),aa_nat_nat(suc,N4)))
                 => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma2),N4) = I2 )
                     => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma2),N4)) )
                    & ! [X: nat] :
                        ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X),N4) = I2 )
                          & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList2,I2)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X),N4)) )
                       => ( 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.simps
tff(fact_8722_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) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X4: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                     => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                 => ( aa_nat_o(vEBT_invar_vebt(Summary),M3)
                   => ( ( 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))),M3) )
                     => ( ( M3 = N2 )
                       => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                         => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_12)
                           => ~ ! [X4: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none_P5556105721700978146at_nat,Deg,TreeList,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X4: vEBT_VEBT] :
                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                       => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                   => ( aa_nat_o(vEBT_invar_vebt(Summary),M3)
                     => ( ( 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))),M3) )
                       => ( ( M3 = aa_nat_nat(suc,N2) )
                         => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                           => ( ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X_12)
                             => ~ ! [X4: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X4: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                     => ( aa_nat_o(vEBT_invar_vebt(Summary),M3)
                       => ( ( 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))),M3) )
                         => ( ( M3 = N2 )
                           => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                             => ( ! [I4: nat] :
                                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                   => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                    <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X4: vEBT_VEBT] :
                                        ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                       => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                     => ~ ( ( Mi != Ma )
                                         => ! [I4: nat] :
                                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                             => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                 => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                & ! [X4: nat] :
                                                    ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I4 )
                                                      & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                                      & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi: nat,Ma: nat] :
                    ( ( A1 = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Deg,TreeList,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X4: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_invar_vebt(X4),N2) )
                       => ( aa_nat_o(vEBT_invar_vebt(Summary),M3)
                         => ( ( 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))),M3) )
                           => ( ( M3 = aa_nat_nat(suc,N2) )
                             => ( ( Deg = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M3) )
                               => ( ! [I4: nat] :
                                      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                     => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),X_13)
                                      <=> aa_nat_o(vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X4: vEBT_VEBT] :
                                          ( member_VEBT_VEBT(X4,set_VEBT_VEBT2(TreeList))
                                         => ~ ? [X_12: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X4),X_12) ) )
                                   => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mi),Ma)
                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Deg))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),M3))
                                               => ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Ma),N2) = I4 )
                                                   => aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Ma),N2)) )
                                                  & ! [X4: nat] :
                                                      ( ( ( aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,X4),N2) = I4 )
                                                        & aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(TreeList,I4)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,X4),N2)) )
                                                     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mi),X4)
                                                        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X4),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_8723_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: 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),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = Nb )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_8724_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list_VEBT_VEBT,Nb: nat,Summarya: vEBT_VEBT,Mb: 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),Mb)
       => ( ( size_s6755466524823107622T_VEBT(TreeLista) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb) )
         => ( ( Mb = aa_nat_nat(suc,Nb) )
           => ( ( Dega = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb) )
             => ( ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X3,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X_1: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X3),X_1) )
                 => aa_nat_o(vEBT_invar_vebt(vEBT_Node(none_P5556105721700978146at_nat,Dega,TreeLista,Summarya)),Dega) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_8725_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list_VEBT_VEBT,Vd: vEBT_VEBT,Xb: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V),TreeLista,Vd)),Xb)
    <=> $let(
          pos: nat,
          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeLista)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeLista,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_8726_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((Uu),(B4)) )
           => ( ( Xaa = zero_zero_nat )
             => ( Y != $ite((B4),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N2: nat] : Xaa = aa_nat_nat(suc,N2)
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz)
             => ( Y != none_nat ) )
           => ( ( ? [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2)
               => ( Y != none_nat ) )
             => ( ( ? [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2)
                 => ( Y != none_nat ) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            some_nat(Mi),
                            $let(
                              l: nat,
                              l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(TreeList)),
                                  $let(
                                    maxlow: option_nat,
                                    maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                    $ite(
                                      ( ( maxlow != none_nat )
                                      & 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(TreeList,h),l)),
                                      $let(
                                        sc2: option_nat,
                                        sc2:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc2 = 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)))))),sc2)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc2))))) ) ) ),
                                  none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_8727_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => $ite(
                Xaa = zero_zero_nat,
                (A4),
                $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb != vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3)
               => $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_8728_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ~ $ite(
                  Xaa = zero_zero_nat,
                  (A4),
                  $ite(Xaa = one_one_nat,(B4),$false) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3)
             => ~ $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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_8729_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero_nat,
                    (A4),
                    $ite(Xaa = one_one_nat,(B4),$false) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw)
           => (Y) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3)
               => ( (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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_8730_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (B4),
                  some_nat(one_one_nat),
                  $ite((A4),some_nat(zero_zero_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat,Ma: nat] :
                ( ? [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz)
               => ( Y != some_nat(Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_8731_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != $ite(
                  (A4),
                  some_nat(zero_zero_nat),
                  $ite((B4),some_nat(one_one_nat),none_nat) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] : Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)
           => ( Y != none_nat ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] : Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz)
               => ( Y != some_nat(Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_8732_prod_Oempty,axiom,
    ! [G: fun_Ex3741802288643809319omplex] : groups4622424608036095791omplex(G,bot_bo7653980558646680370d_enat) = one_one_complex ).

% prod.empty
tff(fact_8733_prod_Oempty,axiom,
    ! [G: fun_Ex1451620480498224165t_real] : groups97031904164794029t_real(G,bot_bo7653980558646680370d_enat) = one_one_real ).

% prod.empty
tff(fact_8734_prod_Oempty,axiom,
    ! [G: fun_Ex4152069839811072081at_rat] : groups2245840878043517529at_rat(G,bot_bo7653980558646680370d_enat) = one_one_rat ).

% prod.empty
tff(fact_8735_prod_Oempty,axiom,
    ! [G: fun_Ex7660298398204468809at_nat] : groups2880970938130013265at_nat(G,bot_bo7653980558646680370d_enat) = one_one_nat ).

% prod.empty
tff(fact_8736_prod_Oempty,axiom,
    ! [G: fun_Ex3482447378695272101at_int] : groups2878480467620962989at_int(G,bot_bo7653980558646680370d_enat) = one_one_int ).

% prod.empty
tff(fact_8737_prod_Oempty,axiom,
    ! [G: fun_real_complex] : groups713298508707869441omplex(G,bot_bot_set_real) = one_one_complex ).

% prod.empty
tff(fact_8738_prod_Oempty,axiom,
    ! [G: fun_real_real] : groups1681761925125756287l_real(G,bot_bot_set_real) = one_one_real ).

% prod.empty
tff(fact_8739_prod_Oempty,axiom,
    ! [G: fun_real_rat] : groups4061424788464935467al_rat(G,bot_bot_set_real) = one_one_rat ).

% prod.empty
tff(fact_8740_prod_Oempty,axiom,
    ! [G: fun_real_nat] : groups4696554848551431203al_nat(G,bot_bot_set_real) = one_one_nat ).

% prod.empty
tff(fact_8741_prod_Oempty,axiom,
    ! [G: fun_real_int] : groups4694064378042380927al_int(G,bot_bot_set_real) = one_one_int ).

% prod.empty
tff(fact_8742_vebt__maxt_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A),(B2))) = $ite(
        (B2),
        some_nat(one_one_nat),
        $ite((A),some_nat(zero_zero_nat),none_nat) ) ).

% vebt_maxt.simps(1)
tff(fact_8743_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : groups708209901874060359at_nat(aTP_Lamm_dg,A3) = one_one_nat ).

% prod.neutral_const
tff(fact_8744_prod_Oneutral__const,axiom,
    ! [A3: set_nat] : groups705719431365010083at_int(aTP_Lamm_dh,A3) = one_one_int ).

% prod.neutral_const
tff(fact_8745_prod_Oneutral__const,axiom,
    ! [A3: set_int] : groups1705073143266064639nt_int(aTP_Lamm_di,A3) = one_one_int ).

% prod.neutral_const
tff(fact_8746_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = groups1705073143266064639nt_int(aTP_Lamm_bc,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,J))) ).

% prod_int_eq
tff(fact_8747_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups705719431365010083at_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J))) = groups1705073143266064639nt_int(aTP_Lamm_bc,set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,I),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),J)))) ).

% prod_int_plus_eq
tff(fact_8748_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex3741802288643809319omplex,A3: set_Extended_enat] :
      ( ( groups4622424608036095791omplex(G,A3) != one_one_complex )
     => ~ ! [A4: extended_enat] :
            ( member_Extended_enat(A4,A3)
           => ( aa_Ext8748583412541189208omplex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8749_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_complex,A3: set_complex] :
      ( ( groups3708469109370488835omplex(G,A3) != one_one_complex )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8750_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_complex,A3: set_real] :
      ( ( groups713298508707869441omplex(G,A3) != one_one_complex )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8751_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_complex,A3: set_nat] :
      ( ( groups6464643781859351333omplex(G,A3) != one_one_complex )
     => ~ ! [A4: nat] :
            ( member_nat(A4,A3)
           => ( aa_nat_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8752_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_complex,A3: set_int] :
      ( ( groups7440179247065528705omplex(G,A3) != one_one_complex )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_complex(G,A4) = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8753_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_Ex1451620480498224165t_real,A3: set_Extended_enat] :
      ( ( groups97031904164794029t_real(G,A3) != one_one_real )
     => ~ ! [A4: extended_enat] :
            ( member_Extended_enat(A4,A3)
           => ( aa_Ext2541572767299617110t_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8754_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_complex_real,A3: set_complex] :
      ( ( groups766887009212190081x_real(G,A3) != one_one_real )
     => ~ ! [A4: complex] :
            ( member_complex(A4,A3)
           => ( aa_complex_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8755_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_real_real,A3: set_real] :
      ( ( groups1681761925125756287l_real(G,A3) != one_one_real )
     => ~ ! [A4: real] :
            ( member_real(A4,A3)
           => ( aa_real_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8756_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_nat_real,A3: set_nat] :
      ( ( groups129246275422532515t_real(G,A3) != one_one_real )
     => ~ ! [A4: nat] :
            ( member_nat(A4,A3)
           => ( aa_nat_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8757_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: fun_int_real,A3: set_int] :
      ( ( groups2316167850115554303t_real(G,A3) != one_one_real )
     => ~ ! [A4: int] :
            ( member_int(A4,A3)
           => ( aa_int_real(G,A4) = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_8758_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_nat] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => ( aa_nat_nat(G,X3) = one_one_nat ) )
     => ( groups708209901874060359at_nat(G,A3) = one_one_nat ) ) ).

% prod.neutral
tff(fact_8759_prod_Oneutral,axiom,
    ! [A3: set_nat,G: fun_nat_int] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => ( aa_nat_int(G,X3) = one_one_int ) )
     => ( groups705719431365010083at_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8760_prod_Oneutral,axiom,
    ! [A3: set_int,G: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( aa_int_int(G,X3) = one_one_int ) )
     => ( groups1705073143266064639nt_int(G,A3) = one_one_int ) ) ).

% prod.neutral
tff(fact_8761_prod_Odistrib,axiom,
    ! [G: fun_nat_nat,H: fun_nat_nat,A3: set_nat] : groups708209901874060359at_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dj(G),H),A3) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,groups708209901874060359at_nat(G,A3)),groups708209901874060359at_nat(H,A3)) ).

% prod.distrib
tff(fact_8762_prod_Odistrib,axiom,
    ! [G: fun_nat_int,H: fun_nat_int,A3: set_nat] : groups705719431365010083at_int(aa_fun5565043122120104496at_int(aTP_Lamm_dk(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups705719431365010083at_int(G,A3)),groups705719431365010083at_int(H,A3)) ).

% prod.distrib
tff(fact_8763_prod_Odistrib,axiom,
    ! [G: fun_int_int,H: fun_int_int,A3: set_int] : groups1705073143266064639nt_int(aa_fun6280345759651049192nt_int(aTP_Lamm_dl(G),H),A3) = aa_int_int(aa_int_fun_int_int(times_times_int,groups1705073143266064639nt_int(G,A3)),groups1705073143266064639nt_int(H,A3)) ).

% prod.distrib
tff(fact_8764_prod__power__distrib,axiom,
    ! [F: fun_nat_nat,A3: set_nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,groups708209901874060359at_nat(F,A3)),Nb) = groups708209901874060359at_nat(aa_nat_fun_nat_nat(aTP_Lamm_dm(F),Nb),A3) ).

% prod_power_distrib
tff(fact_8765_prod__power__distrib,axiom,
    ! [F: fun_nat_int,A3: set_nat,Nb: nat] : aa_nat_int(power_power_int(groups705719431365010083at_int(F,A3)),Nb) = groups705719431365010083at_int(aa_nat_fun_nat_int(aTP_Lamm_dn(F),Nb),A3) ).

% prod_power_distrib
tff(fact_8766_prod__power__distrib,axiom,
    ! [F: fun_int_int,A3: set_int,Nb: nat] : aa_nat_int(power_power_int(groups1705073143266064639nt_int(F,A3)),Nb) = groups1705073143266064639nt_int(aa_nat_fun_int_int(aTP_Lamm_do(F),Nb),A3) ).

% prod_power_distrib
tff(fact_8767_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),groups708209901874060359at_nat(F,A3)) ) ).

% prod_nonneg
tff(fact_8768_prod__nonneg,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8769_prod__nonneg,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_nonneg
tff(fact_8770_prod__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real,G: fun_Ex1451620480498224165t_real] :
      ( ! [I3: extended_enat] :
          ( member_Extended_enat(I3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,I3)),aa_Ext2541572767299617110t_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups97031904164794029t_real(F,A3)),groups97031904164794029t_real(G,A3)) ) ).

% prod_mono
tff(fact_8771_prod__mono,axiom,
    ! [A3: set_complex,F: fun_complex_real,G: fun_complex_real] :
      ( ! [I3: complex] :
          ( member_complex(I3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(F,I3)),aa_complex_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),groups766887009212190081x_real(G,A3)) ) ).

% prod_mono
tff(fact_8772_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_real,G: fun_real_real] :
      ( ! [I3: real] :
          ( member_real(I3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,I3)),aa_real_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),groups1681761925125756287l_real(G,A3)) ) ).

% prod_mono
tff(fact_8773_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_real,G: fun_nat_real] :
      ( ! [I3: nat] :
          ( member_nat(I3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),aa_nat_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A3)),groups129246275422532515t_real(G,A3)) ) ).

% prod_mono
tff(fact_8774_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_real,G: fun_int_real] :
      ( ! [I3: int] :
          ( member_int(I3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_int_real(F,I3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(F,I3)),aa_int_real(G,I3)) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),groups2316167850115554303t_real(G,A3)) ) ).

% prod_mono
tff(fact_8775_prod__mono,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat,G: fun_Ex4152069839811072081at_rat] :
      ( ! [I3: extended_enat] :
          ( member_Extended_enat(I3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Extended_enat_rat(F,I3)),aa_Extended_enat_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2245840878043517529at_rat(F,A3)),groups2245840878043517529at_rat(G,A3)) ) ).

% prod_mono
tff(fact_8776_prod__mono,axiom,
    ! [A3: set_complex,F: fun_complex_rat,G: fun_complex_rat] :
      ( ! [I3: complex] :
          ( member_complex(I3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_complex_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_complex_rat(F,I3)),aa_complex_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),groups225925009352817453ex_rat(G,A3)) ) ).

% prod_mono
tff(fact_8777_prod__mono,axiom,
    ! [A3: set_real,F: fun_real_rat,G: fun_real_rat] :
      ( ! [I3: real] :
          ( member_real(I3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_real_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_real_rat(F,I3)),aa_real_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),groups4061424788464935467al_rat(G,A3)) ) ).

% prod_mono
tff(fact_8778_prod__mono,axiom,
    ! [A3: set_nat,F: fun_nat_rat,G: fun_nat_rat] :
      ( ! [I3: nat] :
          ( member_nat(I3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_nat_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(F,I3)),aa_nat_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),groups73079841787564623at_rat(G,A3)) ) ).

% prod_mono
tff(fact_8779_prod__mono,axiom,
    ! [A3: set_int,F: fun_int_rat,G: fun_int_rat] :
      ( ! [I3: int] :
          ( member_int(I3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(F,I3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(F,I3)),aa_int_rat(G,I3)) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),groups1072433553688619179nt_rat(G,A3)) ) ).

% prod_mono
tff(fact_8780_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_nat] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(F,X3)) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),groups708209901874060359at_nat(F,A3)) ) ).

% prod_pos
tff(fact_8781_prod__pos,axiom,
    ! [A3: set_nat,F: fun_nat_int] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_nat_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups705719431365010083at_int(F,A3)) ) ).

% prod_pos
tff(fact_8782_prod__pos,axiom,
    ! [A3: set_int,F: fun_int_int] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(F,X3)) )
     => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),groups1705073143266064639nt_int(F,A3)) ) ).

% prod_pos
tff(fact_8783_prod__ge__1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X3: extended_enat] :
          ( member_Extended_enat(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_Ext2541572767299617110t_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups97031904164794029t_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8784_prod__ge__1,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_complex_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups766887009212190081x_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8785_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups1681761925125756287l_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8786_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_nat_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups129246275422532515t_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8787_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_int_real(F,X3)) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),groups2316167850115554303t_real(F,A3)) ) ).

% prod_ge_1
tff(fact_8788_prod__ge__1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( ! [X3: extended_enat] :
          ( member_Extended_enat(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_Extended_enat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups2245840878043517529at_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8789_prod__ge__1,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_complex_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups225925009352817453ex_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8790_prod__ge__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_real_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups4061424788464935467al_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8791_prod__ge__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_nat_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups73079841787564623at_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8792_prod__ge__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(F,X3)) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),groups1072433553688619179nt_rat(F,A3)) ) ).

% prod_ge_1
tff(fact_8793_prod__le__1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex1451620480498224165t_real] :
      ( ! [X3: extended_enat] :
          ( member_Extended_enat(X3,A3)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_Ext2541572767299617110t_real(F,X3))
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_Ext2541572767299617110t_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups97031904164794029t_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8794_prod__le__1,axiom,
    ! [A3: set_complex,F: fun_complex_real] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( 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,aa_complex_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups766887009212190081x_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8795_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => ( 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,aa_real_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups1681761925125756287l_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8796_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_real] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => ( 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,aa_nat_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups129246275422532515t_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8797_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_real] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( 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,aa_int_real(F,X3)),one_one_real) ) )
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups2316167850115554303t_real(F,A3)),one_one_real) ) ).

% prod_le_1
tff(fact_8798_prod__le__1,axiom,
    ! [A3: set_Extended_enat,F: fun_Ex4152069839811072081at_rat] :
      ( ! [X3: extended_enat] :
          ( member_Extended_enat(X3,A3)
         => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_Extended_enat_rat(F,X3))
            & aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_Extended_enat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups2245840878043517529at_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8799_prod__le__1,axiom,
    ! [A3: set_complex,F: fun_complex_rat] :
      ( ! [X3: complex] :
          ( member_complex(X3,A3)
         => ( 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,aa_complex_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups225925009352817453ex_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8800_prod__le__1,axiom,
    ! [A3: set_real,F: fun_real_rat] :
      ( ! [X3: real] :
          ( member_real(X3,A3)
         => ( 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,aa_real_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups4061424788464935467al_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8801_prod__le__1,axiom,
    ! [A3: set_nat,F: fun_nat_rat] :
      ( ! [X3: nat] :
          ( member_nat(X3,A3)
         => ( 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,aa_nat_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups73079841787564623at_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8802_prod__le__1,axiom,
    ! [A3: set_int,F: fun_int_rat] :
      ( ! [X3: int] :
          ( member_int(X3,A3)
         => ( 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,aa_int_rat(F,X3)),one_one_rat) ) )
     => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,groups1072433553688619179nt_rat(F,A3)),one_one_rat) ) ).

% prod_le_1
tff(fact_8803_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A),(B2))) = $ite(
        (A),
        some_nat(zero_zero_nat),
        $ite((B2),some_nat(one_one_nat),none_nat) ) ).

% vebt_mint.simps(1)
tff(fact_8804_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option_nat] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(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((Uu),(B4)),zero_zero_nat)) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa_nat_nat(suc,N2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uv),(Uw)),aa_nat_nat(suc,N2))) ) ) )
           => ( ! [Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz) )
                 => ( ( Y = none_nat )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Ux,Uy2,Uz),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2) )
                   => ( ( Y = none_nat )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Vc2,Vd2),Xaa)) ) )
               => ( ! [V3: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),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(V3),aa_nat_nat(suc,zero_zero_nat),Vg2,Vh2),Xaa)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                some_nat(Mi),
                                $let(
                                  l: nat,
                                  l:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,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(TreeList)),
                                      $let(
                                        maxlow: option_nat,
                                        maxlow:= vEBT_vebt_maxt(nth_VEBT_VEBT(TreeList,h)),
                                        $ite(
                                          ( ( maxlow != none_nat )
                                          & 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(TreeList,h),l)),
                                          $let(
                                            sc2: option_nat,
                                            sc2:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc2 = 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)))))),sc2)),vEBT_vebt_mint(nth_VEBT_VEBT(TreeList,the_nat(sc2))))) ) ) ),
                                      none_nat ) ) ) ) )
                         => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_succ_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_8805_ln__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
       => ( aa_real_real(ln_ln_real,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_dp,Xb)) ) ) ) ).

% ln_series
tff(fact_8806_arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => ( aa_real_real(arctan,Xb) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_dq,Xb)) ) ) ).

% arctan_series
tff(fact_8807_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_8808_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_8809_divmod__algorithm__code_I7_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_eq_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit0,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit0,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(7)
tff(fact_8810_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(Mb,one) = product_Pair_int_int(aa_num_int(numeral_numeral_int,Mb),zero_zero_int) ).

% divmod_algorithm_code(2)
tff(fact_8811_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique5055182867167087721od_nat(Mb,one) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_num_nat(numeral_numeral_nat,Mb)),zero_zero_nat) ).

% divmod_algorithm_code(2)
tff(fact_8812_divmod__algorithm__code_I2_J,axiom,
    ! [Mb: num] : unique3479559517661332726nteger(Mb,one) = produc1086072967326762835nteger(numera6620942414471956472nteger(Mb),zero_z3403309356797280102nteger) ).

% divmod_algorithm_code(2)
tff(fact_8813_powser__zero,axiom,
    ! [F: fun_nat_complex] : suminf_complex(aTP_Lamm_dr(F)) = aa_nat_complex(F,zero_zero_nat) ).

% powser_zero
tff(fact_8814_powser__zero,axiom,
    ! [F: fun_nat_real] : suminf_real(aTP_Lamm_ds(F)) = aa_nat_real(F,zero_zero_nat) ).

% powser_zero
tff(fact_8815_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_8816_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_8817_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_8818_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_8819_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_8820_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_8821_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Mb))),unique5026877609467782581ep_nat(aa_num_num(bit1,Nb),unique5055182867167087721od_nat(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_8822_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),unique5024387138958732305ep_int(aa_num_num(bit1,Nb),unique5052692396658037445od_int(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_8823_divmod__algorithm__code_I8_J,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(aa_num_num(bit1,Mb))),unique4921790084139445826nteger(aa_num_num(bit1,Nb),unique3479559517661332726nteger(aa_num_num(bit1,Mb),aa_num_num(bit0,aa_num_num(bit1,Nb))))) ).

% divmod_algorithm_code(8)
tff(fact_8824_divmod__int__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_int_def
tff(fact_8825_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5052692396658037445od_int(Mb,Nb) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))) ).

% divmod_def
tff(fact_8826_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod_def
tff(fact_8827_divmod__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% divmod_def
tff(fact_8828_divmod_H__nat__def,axiom,
    ! [Mb: num,Nb: num] : unique5055182867167087721od_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))) ).

% divmod'_nat_def
tff(fact_8829_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5055182867167087721od_nat(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),aa_num_nat(numeral_numeral_nat,Mb)),unique5026877609467782581ep_nat(Nb,unique5055182867167087721od_nat(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_8830_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique5052692396658037445od_int(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),product_Pair_int_int(zero_zero_int,aa_num_int(numeral_numeral_int,Mb)),unique5024387138958732305ep_int(Nb,unique5052692396658037445od_int(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_8831_divmod__divmod__step,axiom,
    ! [Mb: num,Nb: num] :
      unique3479559517661332726nteger(Mb,Nb) = $ite(aa_num_o(aa_num_fun_num_o(ord_less_num,Mb),Nb),produc1086072967326762835nteger(zero_z3403309356797280102nteger,numera6620942414471956472nteger(Mb)),unique4921790084139445826nteger(Nb,unique3479559517661332726nteger(Mb,aa_num_num(bit0,Nb)))) ).

% divmod_divmod_step
tff(fact_8832_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_dt) ).

% pi_series
tff(fact_8833_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz),Xaa)) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                    $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_8834_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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)) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw),Xaa)) ) )
           => ( ! [V3: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz) )
                 => ( ~ (Y)
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),zero_zero_nat,Uy2,Uz),Xaa)) ) )
             => ( ! [V3: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(V3),aa_nat_nat(suc,zero_zero_nat),Vb2,Vc2),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                      $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_8835_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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)) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3) )
                 => ( ( (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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_8836_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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,V3: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_8837_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_vebt_member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_vebt_member_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Ma),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_vebt_member(nth_VEBT_VEBT(TreeList,h)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_8838_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_V5719532721284313246member(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu,zero_zero_nat,Uv,Uw) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uu,zero_zero_nat,Uv,Uw),Xaa)) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V3: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V5765760719290551771er_rel,produc738532404422230701BT_nat(vEBT_Node(Uy2,aa_nat_nat(suc,V3),TreeList,S3),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,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_V5719532721284313246member(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_8839_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ~ (Y)
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ( ~ (Y)
                 => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2),Xaa)) ) )
               => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                              $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_8840_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) )
         => ( ! [Ux: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,zero_zero_nat,Ux,Uy2),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2) )
                   => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2) )
                     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2),Xaa))
                       => $let(
                            pos: nat,
                            pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_8841_suminf__geometric,axiom,
    ! [C: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(real_V7735802525324610683m_real,C)),one_one_real)
     => ( suminf_real(aa_real_fun_nat_real(power_power_real,C)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),C)) ) ) ).

% suminf_geometric
tff(fact_8842_suminf__geometric,axiom,
    ! [C: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real)
     => ( suminf_complex(power_power_complex(C)) = aa_complex_complex(divide1717551699836669952omplex(one_one_complex),aa_complex_complex(minus_minus_complex(one_one_complex),C)) ) ) ).

% suminf_geometric
tff(fact_8843_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_membermima(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2) )
             => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),zero_zero_nat,Va3,Vb2),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2) )
               => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,V3),TreeList,Vc2),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2) )
                 => ( accp_P2887432264394892906BT_nat(vEBT_V4351362008482014158ma_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,V3),TreeList,Vd2),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,pos),size_s6755466524823107622T_VEBT(TreeList)),aa_nat_o(vEBT_VEBT_membermima(nth_VEBT_VEBT(TreeList,pos)),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,V3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_8844_minus__one__div__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% minus_one_div_numeral
tff(fact_8845_one__div__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,Nb))) ).

% one_div_minus_numeral
tff(fact_8846_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb))
    <=> unique6319869463603278526ux_int(unique5052692396658037445od_int(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_8847_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))
    <=> aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,unique5055182867167087721od_nat(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_8848_dvd__numeral__simp,axiom,
    ! [Mb: num,Nb: num] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,numera6620942414471956472nteger(Mb)),numera6620942414471956472nteger(Nb))
    <=> unique5706413561485394159nteger(unique3479559517661332726nteger(Nb,Mb)) ) ).

% dvd_numeral_simp
tff(fact_8849_minus__numeral__div__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(Mb,Nb))) ).

% minus_numeral_div_numeral
tff(fact_8850_divides__aux__eq,axiom,
    ! [Q2: nat,R2: nat] :
      ( aa_Pro6150637604445964947_nat_o(unique6322359934112328802ux_nat,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Q2),R2))
    <=> ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
tff(fact_8851_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_8852_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(product_Pair_int_int(Q2,R2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Q2),aa_o_int(zero_n2684676970156552555ol_int,R2 != zero_zero_int)) ).

% Divides.adjust_div_eq
tff(fact_8853_numeral__div__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(Mb,Nb))) ).

% numeral_div_minus_numeral
tff(fact_8854_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),zero_zero_int)
     => ( eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B2,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2),product_Pair_int_int(Q2,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_8855_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => summable_real(aa_real_fun_nat_real(aTP_Lamm_dq,Xb)) ) ).

% summable_arctan_series
tff(fact_8856_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B2)
     => ( eucl_rel_int(A,B2,product_Pair_int_int(Q2,R2))
       => eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),B2),product_Pair_int_int(Q2,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_8857_product__nth,axiom,
    ! [Nb: nat,Xs: list_num,Ys2: 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(Ys2)))
     => ( nth_Pr6456567536196504476um_num(product_num_num(Xs,Ys2),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(Ys2))),aa_nat_num(nth_num(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys2)))) ) ) ).

% product_nth
tff(fact_8858_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: 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(Ys2)))
     => ( nth_Pr4953567300277697838T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2),Nb) = produc537772716801021591T_VEBT(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_8859_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: 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(Ys2)))
     => ( nth_Pr4606735188037164562VEBT_o(product_VEBT_VEBT_o(Xs,Ys2),Nb) = produc8721562602347293563VEBT_o(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_o(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_8860_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: 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(Ys2)))
     => ( nth_Pr1791586995822124652BT_nat(produc7295137177222721919BT_nat(Xs,Ys2),Nb) = produc738532404422230701BT_nat(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_nat(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_8861_product__nth,axiom,
    ! [Nb: nat,Xs: list_VEBT_VEBT,Ys2: 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(Ys2)))
     => ( nth_Pr6837108013167703752BT_int(produc7292646706713671643BT_int(Xs,Ys2),Nb) = produc736041933913180425BT_int(nth_VEBT_VEBT(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_int(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_8862_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys2: 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(Ys2)))
     => ( nth_Pr6777367263587873994T_VEBT(product_o_VEBT_VEBT(Xs,Ys2),Nb) = produc2982872950893828659T_VEBT(aa_nat_o(nth_o(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_s6755466524823107622T_VEBT(Ys2))),nth_VEBT_VEBT(Ys2,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_s6755466524823107622T_VEBT(Ys2)))) ) ) ).

% product_nth
tff(fact_8863_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys2: 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(Ys2)))
     => ( nth_Product_prod_o_o(product_o_o(Xs,Ys2),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(Ys2))),aa_nat_o(nth_o(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_o(Ys2)))) ) ) ).

% product_nth
tff(fact_8864_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys2: 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(Ys2)))
     => ( nth_Pr5826913651314560976_o_nat(product_o_nat(Xs,Ys2),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(Ys2))),aa_nat_nat(nth_nat(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_nat(Ys2)))) ) ) ).

% product_nth
tff(fact_8865_product__nth,axiom,
    ! [Nb: nat,Xs: list_o,Ys2: 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(Ys2)))
     => ( nth_Pr1649062631805364268_o_int(product_o_int(Xs,Ys2),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(Ys2))),aa_nat_int(nth_int(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_int(Ys2)))) ) ) ).

% product_nth
tff(fact_8866_product__nth,axiom,
    ! [Nb: nat,Xs: list_nat,Ys2: 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_nat(Xs)),size_size_list_num(Ys2)))
     => ( nth_Pr8326237132889035090at_num(product_nat_num(Xs,Ys2),Nb) = product_Pair_nat_num(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),size_size_list_num(Ys2))),aa_nat_num(nth_num(Ys2),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),size_size_list_num(Ys2)))) ) ) ).

% product_nth
tff(fact_8867_summable__iff__shift,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(aa_nat_fun_nat_real(aTP_Lamm_du(F),K))
    <=> summable_real(F) ) ).

% summable_iff_shift
tff(fact_8868_summable__cmult__iff,axiom,
    ! [C: real,F: fun_nat_real] :
      ( summable_real(aa_fun1839206346889086768t_real(aTP_Lamm_dv(C),F))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_cmult_iff
tff(fact_8869_summable__divide__iff,axiom,
    ! [F: fun_nat_complex,C: complex] :
      ( summable_complex(aa_com8181001469867462503omplex(aTP_Lamm_dw(F),C))
    <=> ( ( C = zero_zero_complex )
        | summable_complex(F) ) ) ).

% summable_divide_iff
tff(fact_8870_summable__divide__iff,axiom,
    ! [F: fun_nat_real,C: real] :
      ( summable_real(aa_real_fun_nat_real(aTP_Lamm_dx(F),C))
    <=> ( ( C = zero_zero_real )
        | summable_real(F) ) ) ).

% summable_divide_iff
tff(fact_8871_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] : size_s7466405169056248089T_VEBT(produc4743750530478302277T_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_s6755466524823107622T_VEBT(Ys2)) ).

% length_product
tff(fact_8872_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_o] : size_s9168528473962070013VEBT_o(product_VEBT_VEBT_o(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_o(Ys2)) ).

% length_product
tff(fact_8873_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_nat] : size_s6152045936467909847BT_nat(produc7295137177222721919BT_nat(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_nat(Ys2)) ).

% length_product
tff(fact_8874_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys2: list_int] : size_s3661962791536183091BT_int(produc7292646706713671643BT_int(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_s6755466524823107622T_VEBT(Xs)),size_size_list_int(Ys2)) ).

% length_product
tff(fact_8875_length__product,axiom,
    ! [Xs: list_o,Ys2: list_VEBT_VEBT] : size_s4313452262239582901T_VEBT(product_o_VEBT_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_s6755466524823107622T_VEBT(Ys2)) ).

% length_product
tff(fact_8876_length__product,axiom,
    ! [Xs: list_o,Ys2: list_o] : size_s1515746228057227161od_o_o(product_o_o(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_o(Ys2)) ).

% length_product
tff(fact_8877_length__product,axiom,
    ! [Xs: list_o,Ys2: list_nat] : size_s5443766701097040955_o_nat(product_o_nat(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_nat(Ys2)) ).

% length_product
tff(fact_8878_length__product,axiom,
    ! [Xs: list_o,Ys2: list_int] : size_s2953683556165314199_o_int(product_o_int(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_o(Xs)),size_size_list_int(Ys2)) ).

% length_product
tff(fact_8879_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_VEBT_VEBT] : size_s4762443039079500285T_VEBT(produc7156399406898700509T_VEBT(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_s6755466524823107622T_VEBT(Ys2)) ).

% length_product
tff(fact_8880_length__product,axiom,
    ! [Xs: list_nat,Ys2: list_o] : size_s6491369823275344609_nat_o(product_nat_o(Xs,Ys2)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_o(Ys2)) ).

% length_product
tff(fact_8881_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( summable_complex(power_power_complex(C))
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V1022390504157884413omplex,C)),one_one_real) ) ).

% summable_geometric_iff
tff(fact_8882_unique__quotient,axiom,
    ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B2,product_Pair_int_int(Q2,R2))
     => ( eucl_rel_int(A,B2,product_Pair_int_int(Q5,R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_8883_unique__remainder,axiom,
    ! [A: int,B2: int,Q2: int,R2: int,Q5: int,R4: int] :
      ( eucl_rel_int(A,B2,product_Pair_int_int(Q2,R2))
     => ( eucl_rel_int(A,B2,product_Pair_int_int(Q5,R4))
       => ( R2 = R4 ) ) ) ).

% unique_remainder
tff(fact_8884_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_8885_summable__rabs__comparison__test,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(F,N2))),aa_nat_real(G,N2)) )
     => ( summable_real(G)
       => summable_real(aTP_Lamm_dy(F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_8886_div__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(aa_int_fun_int_int(divide_divide_int,K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_8887_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(aa_int_fun_int_int(modulo_modulo_int,K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_8888_summable__rabs,axiom,
    ! [F: fun_nat_real] :
      ( summable_real(aTP_Lamm_dy(F))
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,suminf_real(F))),suminf_real(aTP_Lamm_dy(F))) ) ).

% summable_rabs
tff(fact_8889_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K = aa_int_int(aa_int_fun_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_8890_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),L),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),L))) ).

% eucl_rel_int
tff(fact_8891_summable__power__series,axiom,
    ! [F: fun_nat_real,Zb: real] :
      ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,I3)),one_one_real)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(F,I3))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Zb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Zb),one_one_real)
           => summable_real(aa_real_fun_nat_real(aTP_Lamm_dz(F),Zb)) ) ) ) ) ).

% summable_power_series
tff(fact_8892_zminus1__lemma,axiom,
    ! [A: int,B2: int,Q2: int,R2: int] :
      ( eucl_rel_int(A,B2,product_Pair_int_int(Q2,R2))
     => ( ( B2 != zero_zero_int )
       => eucl_rel_int(aa_int_int(uminus_uminus_int,A),B2,
            product_Pair_int_int($ite(R2 = zero_zero_int,aa_int_int(uminus_uminus_int,Q2),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,Q2)),one_one_int)),
              $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B2),R2)))) ) ) ).

% zminus1_lemma
tff(fact_8893_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_8894_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_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_8895_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod_int_int] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero_int )
            & ( A32 = product_Pair_int_int(zero_zero_int,K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,zero_zero_int) )
            & ( L4 != zero_zero_int )
            & ( K3 = aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4) ) )
        | ? [R5: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = product_Pair_int_int(Q4,R5) )
            & ( aa_int_int(sgn_sgn_int,R5) = aa_int_int(sgn_sgn_int,L4) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R5)),aa_int_int(abs_abs_int,L4))
            & ( K3 = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q4),L4)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_8896_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(aa_int_fun_int_int(times_times_int,Q3),A22) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = product_Pair_int_int(Q3,R3) )
               => ( ( aa_int_int(sgn_sgn_int,R3) = aa_int_int(sgn_sgn_int,A22) )
                 => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,R3)),aa_int_int(abs_abs_int,A22))
                   => ( A1 != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Q3),A22)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_8897_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.elims
tff(fact_8898_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se725231765392027082nd_int(K),L) = $ite(
        ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            aa_o_int(zero_n2684676970156552555ol_int,
              ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ).

% and_int.simps
tff(fact_8899_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xb = 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_8900_sin__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ea,Xb)),aa_real_real(sin_real,Xb)) ).

% sin_paired
tff(fact_8901_atLeastAtMostPlus1__int__conv,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb))
     => ( set_or1266510415728281911st_int(Mb,aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb)) = insert_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),Nb),set_or1266510415728281911st_int(Mb,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_8902_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1266510415728281911st_int(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),bot_bot_set_int,insert_int(I,set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J))) ).

% simp_from_to
tff(fact_8903_power__half__series,axiom,
    aa_real_o(sums_real(aTP_Lamm_eb),one_one_real) ).

% power_half_series
tff(fact_8904_sums__if_H,axiom,
    ! [G: fun_nat_real,Xb: real] :
      ( aa_real_o(sums_real(G),Xb)
     => aa_real_o(sums_real(aTP_Lamm_ec(G)),Xb) ) ).

% sums_if'
tff(fact_8905_sums__if,axiom,
    ! [G: fun_nat_real,Xb: real,F: fun_nat_real,Y: real] :
      ( aa_real_o(sums_real(G),Xb)
     => ( aa_real_o(sums_real(F),Y)
       => aa_real_o(sums_real(aa_fun1839206346889086768t_real(aTP_Lamm_ed(G),F)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% sums_if
tff(fact_8906_cos__paired,axiom,
    ! [Xb: real] : aa_real_o(sums_real(aa_real_fun_nat_real(aTP_Lamm_ee,Xb)),aa_real_real(cos_real,Xb)) ).

% cos_paired
tff(fact_8907_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_8908_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa_int_int(bit_se725231765392027082nd_int(Xb),Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xb,Xaa))
       => ~ ( ( Y = $ite(
                  ( member_int(Xb,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int(Xaa,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    aa_o_int(zero_n2684676970156552555ol_int,
                      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      aa_o_int(zero_n2684676970156552555ol_int,
                        ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xb)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Xaa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xb),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) )
           => ~ accp_P1096762738010456898nt_int(bit_and_int_rel,product_Pair_int_int(Xb,Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_8909_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(
            ( member_int(K,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int(L,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              aa_o_int(zero_n2684676970156552555ol_int,
                ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                aa_o_int(zero_n2684676970156552555ol_int,
                  ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ) ) ) ).

% and_int.psimps
tff(fact_8910_obtain__set__succ,axiom,
    ! [Xb: nat,Zb: nat,A3: set_nat,B3: set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Zb)
     => ( aa_nat_o(vEBT_VEBT_max_in_set(A3),Zb)
       => ( finite_finite_nat(B3)
         => ( ( A3 = B3 )
           => ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(A3),Xb),X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_8911_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( aa_nat_o(vEBT_invar_vebt(Ta),Nb)
     => finite_finite_nat(vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_8912_succ__none__empty,axiom,
    ! [Xs: set_nat,A: nat] :
      ( ~ ? [X_1: nat] : aa_nat_o(aa_nat_fun_nat_o(vEBT_is_succ_in_set(Xs),A),X_1)
     => ( finite_finite_nat(Xs)
       => ~ ? [X4: nat] :
              ( member_nat(X4,Xs)
              & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X4) ) ) ) ).

% succ_none_empty
tff(fact_8913_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_finite_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) ).

% finite_atLeastAtMost
tff(fact_8914_finite__nat__set__iff__bounded,axiom,
    ! [N: set_nat] :
      ( finite_finite_nat(N)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( member_nat(X,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_8915_bounded__nat__set__is__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( ! [X3: nat] :
          ( member_nat(X3,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),Nb) )
     => finite_finite_nat(N) ) ).

% bounded_nat_set_is_finite
tff(fact_8916_finite__nat__set__iff__bounded__le,axiom,
    ! [N: set_nat] :
      ( finite_finite_nat(N)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( member_nat(X,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_8917_finite__M__bounded__by__nat,axiom,
    ! [P: fun_nat_o,I: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ef(P),I))) ).

% finite_M_bounded_by_nat
tff(fact_8918_finite__less__ub,axiom,
    ! [F: fun_nat_nat,U: nat] :
      ( ! [N2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N2),aa_nat_nat(F,N2))
     => finite_finite_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eg(F),U))) ) ).

% finite_less_ub
tff(fact_8919_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_8920_Icc__eq__insert__lb__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) = insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_8921_atLeastAtMostSuc__conv,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_nat_nat(suc,Nb))
     => ( aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),aa_nat_nat(suc,Nb)) = insert_nat(aa_nat_nat(suc,Nb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_8922_atLeastAtMost__insertL,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( insert_nat(Mb,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,Mb)),Nb)) = aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_8923_finite__divisors__nat,axiom,
    ! [Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => finite_finite_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ah,Mb))) ) ).

% finite_divisors_nat
tff(fact_8924_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( ord_less_eq_set_nat(N,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),Nb))
     => finite_finite_nat(N) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_8925_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Zb: nat] :
      ( ~ member_nat(Nb,aa_nat_set_nat(nat_set_decode,Zb))
     => ( 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)),Zb)) = insert_nat(Nb,aa_nat_set_nat(nat_set_decode,Zb)) ) ) ).

% set_decode_plus_power_2
tff(fact_8926_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))
           => ( ( ~ ( member_int(K2,insert_int(zero_zero_int,insert_int(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int(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(aa_int_fun_int_int(divide_divide_int,K2),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,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_8927_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eh,K))) ).

% finite_Collect_le_nat
tff(fact_8928_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ei,K))) ).

% finite_Collect_less_nat
tff(fact_8929_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_finite_int(set_or1266510415728281911st_int(L,U)) ).

% finite_atLeastAtMost_int
tff(fact_8930_finite__interval__int4,axiom,
    ! [A: int,B2: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ej(A),B2))) ).

% finite_interval_int4
tff(fact_8931_finite__interval__int3,axiom,
    ! [A: int,B2: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_ek(A),B2))) ).

% finite_interval_int3
tff(fact_8932_finite__interval__int2,axiom,
    ! [A: int,B2: int] : finite_finite_int(collect_int(aa_int_fun_int_o(aTP_Lamm_el(A),B2))) ).

% finite_interval_int2
tff(fact_8933_finite__nth__roots,axiom,
    ! [Nb: nat,C: complex] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => finite3207457112153483333omplex(collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_em(Nb),C))) ) ).

% finite_nth_roots
tff(fact_8934_set__encode__insert,axiom,
    ! [A3: set_nat,Nb: nat] :
      ( finite_finite_nat(A3)
     => ( ~ member_nat(Nb,A3)
       => ( aa_set_nat_nat(nat_set_encode,insert_nat(Nb,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)),aa_set_nat_nat(nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_8935_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun_int_fun_int_o] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I3,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(P,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(P,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_8936_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_8937_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_ei,Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_8938_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bb),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_8939_arith__series__nat,axiom,
    ! [A: nat,D: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_en(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_8940_Sum__Icc__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bb),aa_nat_set_nat(set_or1269000886237332187st_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Icc_nat
tff(fact_8941_even__set__encode__iff,axiom,
    ! [A3: set_nat] :
      ( finite_finite_nat(A3)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_set_nat_nat(nat_set_encode,A3))
      <=> ~ member_nat(zero_zero_nat,A3) ) ) ).

% even_set_encode_iff
tff(fact_8942_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),zero_zero_real)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eo,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_8943_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb)
            & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eo,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_8944_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb)
            & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ep,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_8945_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),Xb)
          & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ep,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_8946_finite__lessThan,axiom,
    ! [K: nat] : finite_finite_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)) ).

% finite_lessThan
tff(fact_8947_lessThan__0,axiom,
    aa_nat_set_nat(set_ord_lessThan_nat,zero_zero_nat) = bot_bot_set_nat ).

% lessThan_0
tff(fact_8948_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : groups6591440286371151544t_real(aTP_Lamm_eq,aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,Nb))) = one_one_real ).

% sumr_cos_zero_one
tff(fact_8949_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_8950_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_8951_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_8952_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_ay,collect_complex(aa_com5808801942499186519plex_o(aTP_Lamm_em(Nb),C))) = zero_zero_complex ) ) ).

% sum_nth_roots
tff(fact_8953_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),Nb)
     => ( groups7754918857620584856omplex(aTP_Lamm_ay,collect_complex(aTP_Lamm_er(Nb))) = zero_zero_complex ) ) ).

% sum_roots_unity
tff(fact_8954_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_es(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_8955_sum__split__even__odd,axiom,
    ! [F: fun_nat_real,G: fun_nat_real,Nb: nat] : groups6591440286371151544t_real(aa_fun1839206346889086768t_real(aTP_Lamm_et(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_eu(F),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),groups6591440286371151544t_real(aTP_Lamm_ev(G),aa_nat_set_nat(set_ord_lessThan_nat,Nb))) ).

% sum_split_even_odd
tff(fact_8956_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ew,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_8957_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(sin_real,Xb)),groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ep,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_8958_Sum__Icc__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Mb),Nb)
     => ( groups4538972089207619220nt_int(aTP_Lamm_bc,set_or1266510415728281911st_int(Mb,Nb)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Nb),aa_int_int(aa_int_fun_int_int(plus_plus_int,Nb),one_one_int))),aa_int_int(aa_int_fun_int_int(times_times_int,Mb),aa_int_int(aa_int_fun_int_int(minus_minus_int,Mb),one_one_int)))),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) ) ) ).

% Sum_Icc_int
tff(fact_8959_sum__pos__lt__pair,axiom,
    ! [F: fun_nat_real,K: nat] :
      ( summable_real(F)
     => ( ! [D2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(suc,aa_nat_nat(suc,zero_zero_nat))),D2)))),aa_nat_real(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,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))),D2)),one_one_nat)))))
       => aa_real_o(aa_real_fun_real_o(ord_less_real,groups6591440286371151544t_real(F,aa_nat_set_nat(set_ord_lessThan_nat,K))),suminf_real(F)) ) ) ).

% sum_pos_lt_pair
tff(fact_8960_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero_real )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
            & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(exp_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ew,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(exp_real,T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_8961_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T: real] : aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ep,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_8962_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(sin_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ep,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(sin_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_8963_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
      & ( aa_real_real(cos_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_eo,Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,T),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,Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_8964_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => bij_betw_nat_complex(aTP_Lamm_ex(Nb),aa_nat_set_nat(set_ord_lessThan_nat,Nb),collect_complex(aTP_Lamm_er(Nb))) ) ).

% bij_betw_roots_unity
tff(fact_8965_finite__atMost,axiom,
    ! [K: nat] : finite_finite_nat(aa_nat_set_nat(set_ord_atMost_nat,K)) ).

% finite_atMost
tff(fact_8966_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_8967_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_8968_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_8969_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_8970_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_8971_sum__choose__upper,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ey,Mb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),aa_nat_nat(suc,Mb)) ).

% sum_choose_upper
tff(fact_8972_sum__choose__lower,axiom,
    ! [R2: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ez,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_8973_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fa,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),Mb) ).

% choose_rising_sum(2)
tff(fact_8974_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fa,Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Mb)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_8975_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_8976_sum__choose__diagonal,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb(Mb),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(suc,Nb)),Mb) ) ) ).

% sum_choose_diagonal
tff(fact_8977_vandermonde,axiom,
    ! [Mb: nat,Nb: nat,R2: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_fc(Mb),Nb),R2)),aa_nat_set_nat(set_ord_atMost_nat,R2)) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),R2) ).

% vandermonde
tff(fact_8978_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_8979_binomial,axiom,
    ! [A: nat,B2: 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),B2)),Nb) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_fd(A),B2),Nb)),aa_nat_set_nat(set_ord_atMost_nat,Nb)) ).

% binomial
tff(fact_8980_polynomial__product__nat,axiom,
    ! [Mb: nat,A: fun_nat_nat,Nb: nat,B2: fun_nat_nat,Xb: nat] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),I3)
         => ( aa_nat_nat(A,I3) = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),J2)
           => ( aa_nat_nat(B2,J2) = zero_zero_nat ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fe(A),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Mb))),aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fe(B2),Xb)),aa_nat_set_nat(set_ord_atMost_nat,Nb))) = aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fg(A),B2),Xb)),aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_8981_choose__square__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fh,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_8982_binomial__r__part__sum,axiom,
    ! [Mb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)),one_one_nat))),aa_nat_set_nat(set_ord_atMost_nat,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)) ).

% binomial_r_part_sum
tff(fact_8983_choose__linear__sum,axiom,
    ! [Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_fi,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_8984_of__nat__id,axiom,
    ! [Nb: nat] : aa_nat_nat(semiri1316708129612266289at_nat,Nb) = Nb ).

% of_nat_id
tff(fact_8985_real__scaleR__def,axiom,
    ! [A: real,Xb: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V1485227260804924795R_real,A),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,A),Xb) ).

% real_scaleR_def
tff(fact_8986_complex__scaleR,axiom,
    ! [R2: real,A: real,B2: real] : aa_complex_complex(real_V2046097035970521341omplex(R2),aa_real_complex(complex2(A),B2)) = 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),B2)) ).

% complex_scaleR
tff(fact_8987_bij__betw__nth__root__unity,axiom,
    ! [C: complex,Nb: nat] :
      ( ( C != zero_zero_complex )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => bij_be1856998921033663316omplex(times_times_complex(aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(aa_nat_fun_real_real(root,Nb),aa_complex_real(real_V1022390504157884413omplex,C)))),aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(arg,C)),aa_nat_real(semiri5074537144036343181t_real,Nb))))),collect_complex(aTP_Lamm_er(Nb)),collect_complex(aa_nat_fun_complex_o(aTP_Lamm_fj(C),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_8988_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ~ finite_finite_int(S2)
    <=> ! [M6: int] :
        ? [N4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,M6),aa_int_int(abs_abs_int,N4))
          & member_int(N4,S2) ) ) ).

% infinite_int_iff_unbounded
tff(fact_8989_Arg__def,axiom,
    ! [Zb: complex] :
      aa_complex_real(arg,Zb) = $ite(Zb = zero_zero_complex,zero_zero_real,fChoice_real(aTP_Lamm_fk(Zb))) ).

% Arg_def
tff(fact_8990_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_8991_real__root__Suc__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(suc,zero_zero_nat)),Xb) = Xb ).

% real_root_Suc_0
tff(fact_8992_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_nat_fun_real_real(root,Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_8993_root__0,axiom,
    ! [Xb: real] : aa_real_real(aa_nat_fun_real_real(root,zero_zero_nat),Xb) = zero_zero_real ).

% root_0
tff(fact_8994_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = zero_zero_real )
      <=> ( Xb = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
tff(fact_8995_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_8996_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_8997_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_8998_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = one_one_real )
      <=> ( Xb = one_one_real ) ) ) ).

% real_root_eq_1_iff
tff(fact_8999_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_9000_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real) ) ) ).

% real_root_lt_0_iff
tff(fact_9001_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),zero_zero_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real) ) ) ).

% real_root_le_0_iff
tff(fact_9002_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_9003_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_9004_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real) ) ) ).

% real_root_lt_1_iff
tff(fact_9005_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_9006_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),one_one_real)
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real) ) ) ).

% real_root_le_1_iff
tff(fact_9007_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_9008_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(uminus_uminus_real,Xb)) = aa_real_real(uminus_uminus_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_minus
tff(fact_9009_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,Xb)) = aa_real_real(inverse_inverse_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_inverse
tff(fact_9010_real__root__commute,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_nat_fun_real_real(root,Mb),Xb)) ).

% real_root_commute
tff(fact_9011_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_mult
tff(fact_9012_real__root__mult__exp,axiom,
    ! [Mb: nat,Nb: nat,Xb: real] : aa_real_real(aa_nat_fun_real_real(root,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),Xb) = aa_real_real(aa_nat_fun_real_real(root,Mb),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ).

% real_root_mult_exp
tff(fact_9013_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ).

% real_root_divide
tff(fact_9014_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_9015_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_9016_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_9017_real__root__power,axiom,
    ! [Nb: nat,Xb: 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,Xb),K)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_9018_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_real_real(abs_abs_real,Xb)) = aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_9019_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) = aa_real_real(sgn_sgn_real,Xb) ) ) ).

% sgn_root
tff(fact_9020_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_9021_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_9022_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_9023_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_9024_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_9025_real__root__strict__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_9026_real__root__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),Xb)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,N),Xb)),aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_9027_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_9028_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Y)
       => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
         => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_9029_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_9030_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_9031_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( aa_nat_real(aa_real_fun_nat_real(power_power_real,Y),Nb) = Xb )
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_9032_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( aa_real_real(aa_nat_fun_real_real(root,Nb),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_9033_real__root__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),N)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_real_real(aa_nat_fun_real_real(root,N),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_9034_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_9035_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_9036_ln__root,axiom,
    ! [Nb: nat,B2: 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),B2)
       => ( aa_real_real(ln_ln_real,aa_real_real(aa_nat_fun_real_real(root,Nb),B2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,B2)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% ln_root
tff(fact_9037_log__root,axiom,
    ! [Nb: nat,A: real,B2: 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,B2),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,B2),A)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ) ) ) ).

% log_root
tff(fact_9038_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),B2)
       => ( aa_real_real(aa_rea7340605660300375871l_real(log,aa_real_real(aa_nat_fun_real_real(root,Nb),B2)),Xb) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_real_real(aa_rea7340605660300375871l_real(log,B2),Xb)) ) ) ) ).

% log_base_root
tff(fact_9039_split__root,axiom,
    ! [P: fun_real_o,Nb: nat,Xb: real] :
      ( aa_real_o(P,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb))
    <=> ( ( ( Nb = zero_zero_nat )
         => aa_real_o(P,zero_zero_real) )
        & ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
         => ! [Y5: real] :
              ( ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Y5)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Y5)),Nb)) = Xb )
             => aa_real_o(P,Y5) ) ) ) ) ).

% split_root
tff(fact_9040_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ~ finite_finite_nat(S2)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),N4)
          & member_nat(N4,S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_9041_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K),M3)
         => ? [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),N5)
              & member_nat(N5,S2) ) )
     => ~ finite_finite_nat(S2) ) ).

% unbounded_k_infinite
tff(fact_9042_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ~ finite_finite_nat(S2)
    <=> ! [M6: nat] :
        ? [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M6),N4)
          & member_nat(N4,S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_9043_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => ( aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_9044_divmod__BitM__2__eq,axiom,
    ! [Mb: num] : unique5052692396658037445od_int(aa_num_num(bitM,Mb),aa_num_num(bit0,one)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_9045_insert__simp__excp,axiom,
    ! [Mia: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mia),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Mia),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_9046_insert__simp__norm,axiom,
    ! [Xb: nat,Dega: nat,TreeLista: list_VEBT_VEBT,Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),size_s6755466524823107622T_VEBT(TreeLista))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mia),Xb)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Dega)
         => ( ( Xb != Maa )
           => ( vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),Dega,TreeLista,Summarya),Xb) = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Xb),Maa))),Dega,list_u1324408373059187874T_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),vEBT_vebt_insert(nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                  $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeLista,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),vEBT_vebt_insert(Summarya,aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,Xb),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_9047_max__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_Suc_Suc
tff(fact_9048_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_9049_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_9050_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_9051_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_9052_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_9053_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B2: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B2) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        & ( B2 = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_9054_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_9055_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_9056_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_9057_nat__add__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)) ).

% nat_add_max_right
tff(fact_9058_nat__add__max__left,axiom,
    ! [Mb: 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,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),Q2)) ).

% nat_add_max_left
tff(fact_9059_nat__mult__max__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_max_right
tff(fact_9060_nat__mult__max__left,axiom,
    ! [Mb: 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,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_max_left
tff(fact_9061_semiring__norm_I26_J,axiom,
    aa_num_num(bitM,one) = one ).

% semiring_norm(26)
tff(fact_9062_nat__minus__add__max,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Nb),Mb) ).

% nat_minus_add_max
tff(fact_9063_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_9064_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_9065_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_9066_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_9067_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_9068_one__plus__BitM,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(one),aa_num_num(bitM,Nb)) = aa_num_num(bit0,Nb) ).

% one_plus_BitM
tff(fact_9069_BitM__plus__one,axiom,
    ! [Nb: num] : aa_num_num(plus_plus_num(aa_num_num(bitM,Nb)),one) = aa_num_num(bit0,Nb) ).

% BitM_plus_one
tff(fact_9070_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Va2: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Mia,Xb),
        $let(
          h: nat,
          h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va2))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
          $ite(
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Mia),Xb,Mia)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Maa))),aa_nat_nat(suc,aa_nat_nat(suc,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(Mia),Maa)),aa_nat_nat(suc,aa_nat_nat(suc,Va2)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_9071_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = 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,S3: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
             => ( Y != vEBT_Node(Info2,zero_zero_nat,Ts2,S3) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
               => ( Y != vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) ) )
           => ( ! [V3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary) )
                 => ( Y != vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                          $let(
                            h: nat,
                            h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                            $ite(
                              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = Ma ) ) ),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_9072_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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,S3: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
               => ( ( Y = vEBT_Node(Info2,zero_zero_nat,Ts2,S3) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,zero_zero_nat,Ts2,S3),Xaa)) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                 => ( ( Y = vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3) )
                   => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(Info2,aa_nat_nat(suc,zero_zero_nat),Ts2,S3),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary) )
                   => ( ( Y = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),Xaa)),aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary) )
                     => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(none_P5556105721700978146at_nat,aa_nat_nat(suc,aa_nat_nat(suc,V3)),TreeList,Summary),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Mi,Xaa),
                              $let(
                                h: nat,
                                h:= aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),
                                $ite(
                                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,h),size_s6755466524823107622T_VEBT(TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = Ma ) ) ),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat($ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xaa),Mi),Xaa,Mi)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,xn),Ma))),aa_nat_nat(suc,aa_nat_nat(suc,Va)),list_u1324408373059187874T_VEBT(TreeList,h,vEBT_vebt_insert(nth_VEBT_VEBT(TreeList,h),aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_low,xn),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(suc,aa_nat_nat(suc,Va))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),
                                    $ite(aa_VEBT_VEBT_o(vEBT_VEBT_minNull,nth_VEBT_VEBT(TreeList,h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary) ) ) ) )
                       => ~ accp_P2887432264394892906BT_nat(vEBT_vebt_insert_rel,produc738532404422230701BT_nat(vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),aa_nat_nat(suc,aa_nat_nat(suc,Va)),TreeList,Summary),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_9073_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),zero_zero_nat) ) )
         => ( ( ( Xb = aa_nat_nat(suc,zero_zero_nat) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa_nat_o(accp_nat(vEBT_v4011308405150292612up_rel),aa_nat_nat(suc,zero_zero_nat)) ) )
           => ~ ! [Va: nat] :
                  ( ( Xb = 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_9074_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_9075_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_9076_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fl(L)),Qr) ).

% divmod_step_nat_def
tff(fact_9077_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_fm(L),Qr) ).

% divmod_step_int_def
tff(fact_9078_divmod__nat__if,axiom,
    ! [Mb: nat,Nb: nat] :
      divmod_nat(Mb,Nb) = $ite(
        ( ( Nb = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ),
        aa_nat7603221163087113075at_nat(product_Pair_nat_nat(zero_zero_nat),Mb),
        aa_Pro262675715343573244at_nat(produc2626176000494625587at_nat(aTP_Lamm_fn),divmod_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_9079_arctan__def,axiom,
    ! [Y: real] : aa_real_real(arctan,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_fo,Y)) ).

% arctan_def
tff(fact_9080_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),zero_zero_real)
     => ( aa_real_real(ln_ln_real,Xb) = the_real(aTP_Lamm_fp) ) ) ).

% ln_neg_is_const
tff(fact_9081_arccos__def,axiom,
    ! [Y: real] : aa_real_real(arccos,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_fq,Y)) ).

% arccos_def
tff(fact_9082_divmod__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : divmod_nat(Mb,Nb) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb)) ).

% divmod_nat_def
tff(fact_9083_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_fr) ).

% pi_half
tff(fact_9084_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_fr)) ).

% pi_def
tff(fact_9085_arcsin__def,axiom,
    ! [Y: real] : aa_real_real(arcsin,Y) = the_real(aa_real_fun_real_o(aTP_Lamm_fs,Y)) ).

% arcsin_def
tff(fact_9086_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(aa_int_fun_int_int(plus_plus_int,aa_int_int(ord_max_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))))) ) ) ).

% or_int_unfold
tff(fact_9087_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_9088_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_9089_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_9090_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_9091_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_9092_or__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se725231765392027082nd_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_9093_and__minus__minus__numerals,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Nb)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_9094_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_9095_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Y)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(bit_se1409905431419307370or_int(Xb),Y)) ) ) ).

% OR_lower
tff(fact_9096_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_9097_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_se725231765392027082nd_int(Xb),Y)),aa_int_int(bit_se1409905431419307370or_int(Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),Y) ).

% plus_and_or
tff(fact_9098_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_9099_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_9100_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_9101_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_9102_set__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(aa_nat_fun_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_9103_or__not__numerals_I4_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_9104_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_9105_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_9106_or__not__numerals_I7_J,axiom,
    ! [Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_9107_or__not__numerals_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% or_not_numerals(6)
tff(fact_9108_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),Xb)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb))
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(bit_se1409905431419307370or_int(Xb),Y)),aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Nb)) ) ) ) ).

% OR_upper
tff(fact_9109_or__not__numerals_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(5)
tff(fact_9110_or__not__numerals_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(9)
tff(fact_9111_or__not__numerals_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,aa_num_num(bit1,Mb))),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))))) ).

% or_not_numerals(8)
tff(fact_9112_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa_int_int(bit_se1409905431419307370or_int(K),L) = aa_int_int(aa_int_fun_int_int(plus_plus_int,
          aa_o_int(zero_n2684676970156552555ol_int,
            ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),K)
            | ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_int_int(bit_se1409905431419307370or_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one)))))) ).

% or_int_rec
tff(fact_9113_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(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(5)
tff(fact_9114_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(bit_or_not_num_neg(one),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(1)
tff(fact_9115_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_9116_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_9117_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_9118_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,Xb))),aa_nat_nat(suc,zero_zero_nat)) = aa_num_nat(numeral_numeral_nat,aa_num_num(bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_9119_or__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(4)
tff(fact_9120_or__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bit0,Nb)))) ).

% or_minus_numerals(8)
tff(fact_9121_or__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(7)
tff(fact_9122_or__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),aa_num_num(bitM,Nb)))) ).

% or_minus_numerals(3)
tff(fact_9123_or__not__num__neg_Osimps_I1_J,axiom,
    aa_num_num(bit_or_not_num_neg(one),one) = one ).

% or_not_num_neg.simps(1)
tff(fact_9124_set__bit__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,Mb),one_one_nat)) ).

% set_bit_nat_def
tff(fact_9125_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : aa_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_9126_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(6)
tff(fact_9127_or__not__num__neg_Osimps_I3_J,axiom,
    ! [Mb: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit1,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(3)
tff(fact_9128_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),one) = one ).

% or_not_num_neg.simps(7)
tff(fact_9129_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit0,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(5)
tff(fact_9130_or__not__num__neg_Osimps_I9_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit1,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(9)
tff(fact_9131_or__nat__def,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_int_nat(nat2,aa_int_int(bit_se1409905431419307370or_int(aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))) ).

% or_nat_def
tff(fact_9132_or__not__num__neg_Osimps_I2_J,axiom,
    ! [Mb: num] : aa_num_num(bit_or_not_num_neg(one),aa_num_num(bit0,Mb)) = aa_num_num(bit1,Mb) ).

% or_not_num_neg.simps(2)
tff(fact_9133_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_num_num(bit_or_not_num_neg(aa_num_num(bit1,Nb)),aa_num_num(bit0,Mb)) = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(Nb),Mb)) ).

% or_not_num_neg.simps(8)
tff(fact_9134_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(bit_or_not_num_neg(Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != one ) ) )
       => ( ( ( Xb = one )
           => ! [M3: num] :
                ( ( Xaa = aa_num_num(bit0,M3) )
               => ( Y != aa_num_num(bit1,M3) ) ) )
         => ( ( ( Xb = one )
             => ! [M3: num] :
                  ( ( Xaa = aa_num_num(bit1,M3) )
                 => ( Y != aa_num_num(bit1,M3) ) ) )
           => ( ( ? [N2: num] : Xb = aa_num_num(bit0,N2)
               => ( ( Xaa = one )
                 => ( Y != aa_num_num(bit0,one) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = aa_num_num(bit0,N2) )
                   => ! [M3: num] :
                        ( ( Xaa = aa_num_num(bit0,M3) )
                       => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = aa_num_num(bit0,N2) )
                     => ! [M3: num] :
                          ( ( Xaa = aa_num_num(bit1,M3) )
                         => ( Y != aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N2),M3)) ) ) )
                 => ( ( ? [N2: num] : Xb = aa_num_num(bit1,N2)
                     => ( ( Xaa = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa_num_num(bit1,N2) )
                         => ! [M3: num] :
                              ( ( Xaa = aa_num_num(bit0,M3) )
                             => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( Xb = aa_num_num(bit1,N2) )
                           => ! [M3: num] :
                                ( ( Xaa = aa_num_num(bit1,M3) )
                               => ( Y != aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_9135_numeral__or__not__num__eq,axiom,
    ! [Mb: num,Nb: num] : aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),Nb)) = aa_int_int(uminus_uminus_int,aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb)))) ).

% numeral_or_not_num_eq
tff(fact_9136_int__numeral__not__or__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Nb),Mb))) ).

% int_numeral_not_or_num_neg
tff(fact_9137_int__numeral__or__not__num__neg,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se1409905431419307370or_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit_or_not_num_neg(Mb),Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_9138_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% or_Suc_0_eq
tff(fact_9139_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Nb),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) ).

% Suc_0_or_eq
tff(fact_9140_or__nat__rec,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,
          zero_n2687167440665602831ol_nat(( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Mb)
            | ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb) ))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% or_nat_rec
tff(fact_9141_or__nat__unfold,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,Mb),Nb) = $ite(
        Mb = zero_zero_nat,
        Nb,
        $ite(Nb = zero_zero_nat,Mb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Nb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))) ) ).

% or_nat_unfold
tff(fact_9142_Sum__Ico__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bb),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),one_one_nat)))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% Sum_Ico_nat
tff(fact_9143_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_9144_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_finite_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) ).

% finite_atLeastLessThan
tff(fact_9145_atLeastLessThan__singleton,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Mb)) = insert_nat(Mb,bot_bot_set_nat) ).

% atLeastLessThan_singleton
tff(fact_9146_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),Nb)
         => aa_nat_o(P,M6) )
    <=> ! [X: nat] :
          ( member_nat(X,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
         => aa_nat_o(P,X) ) ) ).

% all_nat_less_eq
tff(fact_9147_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun_nat_o] :
      ( ? [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M6),Nb)
          & aa_nat_o(P,M6) )
    <=> ? [X: nat] :
          ( member_nat(X,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
          & aa_nat_o(P,X) ) ) ).

% ex_nat_less_eq
tff(fact_9148_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_9149_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_9150_atLeastLessThan0,axiom,
    ! [Mb: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),zero_zero_nat) = bot_bot_set_nat ).

% atLeastLessThan0
tff(fact_9151_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_9152_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N: set_nat,Nb: nat] :
      ( ord_less_eq_set_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => finite_finite_nat(N) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_9153_atLeastLessThanSuc,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_nat_nat(suc,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb),insert_nat(Nb,aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)),bot_bot_set_nat) ).

% atLeastLessThanSuc
tff(fact_9154_prod__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_fact
tff(fact_9155_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : groups708209901874060359at_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,zero_zero_nat)),Nb)) = aa_nat_nat(semiri1408675320244567234ct_nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_9156_atLeastLessThan__nat__numeral,axiom,
    ! [Mb: nat,K: num] :
      aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(numeral_numeral_nat,K)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),aa_num_nat(pred_numeral,K)),insert_nat(aa_num_nat(pred_numeral,K),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),aa_num_nat(pred_numeral,K))),bot_bot_set_nat) ).

% atLeastLessThan_nat_numeral
tff(fact_9157_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_9158_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A: fun_nat_nat,B2: fun_nat_nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
         => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(A,I3)),aa_nat_nat(A,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),J2)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,J2),Nb)
             => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(B2,J2)),aa_nat_nat(B2,I3)) ) )
       => 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_dj(A),B2)),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(B2),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_9159_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_finite_int(set_or4662586982721622107an_int(L,U)) ).

% finite_atLeastLessThan_int
tff(fact_9160_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite_int(set_or4662586982721622107an_int(zero_zero_int,U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_9161_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(L,aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = set_or1266510415728281911st_int(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_9162_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_9163_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_9164_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_9165_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(vEBT_Leaf((Uu2),(Uv2))),D)
    <=> ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_9166_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_ft(L),Qr) ).

% divmod_step_integer_def
tff(fact_9167_csqrt_Osimps_I1_J,axiom,
    ! [Zb: complex] : aa_complex_real(re,aa_complex_complex(csqrt,Zb)) = 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,Zb)),aa_complex_real(re,Zb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% csqrt.simps(1)
tff(fact_9168_complex__Re__numeral,axiom,
    ! [V: num] : aa_complex_real(re,numera6690914467698888265omplex(V)) = aa_num_real(numeral_numeral_real,V) ).

% complex_Re_numeral
tff(fact_9169_Re__divide__of__nat,axiom,
    ! [Zb: complex,Nb: nat] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Zb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Zb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Re_divide_of_nat
tff(fact_9170_Re__divide__of__real,axiom,
    ! [Zb: complex,R2: real] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Zb),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Zb)),R2) ).

% Re_divide_of_real
tff(fact_9171_Re__sgn,axiom,
    ! [Zb: complex] : aa_complex_real(re,aa_complex_complex(sgn_sgn_complex,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Zb)),aa_complex_real(real_V1022390504157884413omplex,Zb)) ).

% Re_sgn
tff(fact_9172_Re__divide__numeral,axiom,
    ! [Zb: complex,W: num] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Zb),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Zb)),aa_num_real(numeral_numeral_real,W)) ).

% Re_divide_numeral
tff(fact_9173_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : plus_p5714425477246183910nteger(K,zero_z3403309356797280102nteger) = K ).

% plus_integer_code(1)
tff(fact_9174_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : plus_p5714425477246183910nteger(zero_z3403309356797280102nteger,L) = L ).

% plus_integer_code(2)
tff(fact_9175_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_9176_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_9177_divmod__integer_H__def,axiom,
    ! [Mb: num,Nb: num] : unique3479559517661332726nteger(Mb,Nb) = produc1086072967326762835nteger(divide6298287555418463151nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb)),modulo364778990260209775nteger(numera6620942414471956472nteger(Mb),numera6620942414471956472nteger(Nb))) ).

% divmod_integer'_def
tff(fact_9178_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_9179_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,L) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,L) ).

% minus_integer_code(2)
tff(fact_9180_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : minus_8373710615458151222nteger(K,zero_z3403309356797280102nteger) = K ).

% minus_integer_code(1)
tff(fact_9181_complex__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% complex_Re_le_cmod
tff(fact_9182_one__complex_Osimps_I1_J,axiom,
    aa_complex_real(re,one_one_complex) = one_one_real ).

% one_complex.simps(1)
tff(fact_9183_plus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y)) ).

% plus_complex.simps(1)
tff(fact_9184_scaleR__complex_Osimps_I1_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(re,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb)) ).

% scaleR_complex.simps(1)
tff(fact_9185_minus__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y)) ).

% minus_complex.simps(1)
tff(fact_9186_abs__Re__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Re_le_cmod
tff(fact_9187_Re__csqrt,axiom,
    ! [Zb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Zb))) ).

% Re_csqrt
tff(fact_9188_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_9189_cmod__plus__Re__le__0__iff,axiom,
    ! [Zb: 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,Zb)),aa_complex_real(re,Zb))),zero_zero_real)
    <=> ( aa_complex_real(re,Zb) = aa_real_real(uminus_uminus_real,aa_complex_real(real_V1022390504157884413omplex,Zb)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_9190_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_9191_csqrt_Ocode,axiom,
    ! [Zb: complex] :
      aa_complex_complex(csqrt,Zb) = 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,Zb)),aa_complex_real(re,Zb))),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,Zb) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Zb)))),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,Zb)),aa_complex_real(re,Zb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))))) ).

% csqrt.code
tff(fact_9192_csqrt_Osimps_I2_J,axiom,
    ! [Zb: complex] :
      aa_complex_real(im,aa_complex_complex(csqrt,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,
          $ite(aa_complex_real(im,Zb) = zero_zero_real,one_one_real,aa_real_real(sgn_sgn_real,aa_complex_real(im,Zb)))),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,Zb)),aa_complex_real(re,Zb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))))) ).

% csqrt.simps(2)
tff(fact_9193_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,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))),code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))) = zero_zero_int,l,plus_p5714425477246183910nteger(l,one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_9194_Im__i__times,axiom,
    ! [Zb: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(imaginary_unit),Zb)) = aa_complex_real(re,Zb) ).

% Im_i_times
tff(fact_9195_Im__divide__of__real,axiom,
    ! [Zb: complex,R2: real] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Zb),aa_real_complex(real_V4546457046886955230omplex,R2))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Zb)),R2) ).

% Im_divide_of_real
tff(fact_9196_Im__sgn,axiom,
    ! [Zb: complex] : aa_complex_real(im,aa_complex_complex(sgn_sgn_complex,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Zb)),aa_complex_real(real_V1022390504157884413omplex,Zb)) ).

% Im_sgn
tff(fact_9197_Re__power__real,axiom,
    ! [Xb: complex,Nb: nat] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),Nb)) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),Nb) ) ) ).

% Re_power_real
tff(fact_9198_Re__i__times,axiom,
    ! [Zb: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(imaginary_unit),Zb)) = aa_real_real(uminus_uminus_real,aa_complex_real(im,Zb)) ).

% Re_i_times
tff(fact_9199_Im__divide__numeral,axiom,
    ! [Zb: complex,W: num] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Zb),numera6690914467698888265omplex(W))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Zb)),aa_num_real(numeral_numeral_real,W)) ).

% Im_divide_numeral
tff(fact_9200_Im__divide__of__nat,axiom,
    ! [Zb: complex,Nb: nat] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Zb),aa_nat_complex(semiri8010041392384452111omplex,Nb))) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Zb)),aa_nat_real(semiri5074537144036343181t_real,Nb)) ).

% Im_divide_of_nat
tff(fact_9201_csqrt__of__real__nonneg,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb))
       => ( aa_complex_complex(csqrt,Xb) = aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_complex_real(re,Xb))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_9202_csqrt__minus,axiom,
    ! [Xb: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,Xb)),zero_zero_real)
        | ( ( aa_complex_real(im,Xb) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(re,Xb)) ) )
     => ( aa_complex_complex(csqrt,aa_complex_complex(uminus1482373934393186551omplex,Xb)) = aa_complex_complex(times_times_complex(imaginary_unit),aa_complex_complex(csqrt,Xb)) ) ) ).

% csqrt_minus
tff(fact_9203_csqrt__of__real__nonpos,axiom,
    ! [Xb: complex] :
      ( ( aa_complex_real(im,Xb) = zero_zero_real )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,Xb)),zero_zero_real)
       => ( aa_complex_complex(csqrt,Xb) = aa_complex_complex(times_times_complex(imaginary_unit),aa_real_complex(real_V4546457046886955230omplex,aa_real_real(sqrt,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_9204_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : divide6298287555418463151nteger(code_integer_of_int(Xaa),code_integer_of_int(Xb)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_9205_modulo__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : modulo364778990260209775nteger(code_integer_of_int(Xaa),code_integer_of_int(Xb)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xaa),Xb)) ).

% modulo_integer.abs_eq
tff(fact_9206_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_9207_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_9208_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_9209_imaginary__unit_Osimps_I2_J,axiom,
    aa_complex_real(im,imaginary_unit) = one_one_real ).

% imaginary_unit.simps(2)
tff(fact_9210_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : plus_p5714425477246183910nteger(code_integer_of_int(Xaa),code_integer_of_int(Xb)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_9211_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(times_times_int,Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_9212_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : minus_8373710615458151222nteger(code_integer_of_int(Xaa),code_integer_of_int(Xb)) = code_integer_of_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_9213_plus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)) ).

% plus_complex.simps(2)
tff(fact_9214_scaleR__complex_Osimps_I2_J,axiom,
    ! [R2: real,Xb: complex] : aa_complex_real(im,aa_complex_complex(real_V2046097035970521341omplex(R2),Xb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb)) ).

% scaleR_complex.simps(2)
tff(fact_9215_minus__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)) ).

% minus_complex.simps(2)
tff(fact_9216_abs__Im__le__cmod,axiom,
    ! [Xb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_complex_real(real_V1022390504157884413omplex,Xb)) ).

% abs_Im_le_cmod
tff(fact_9217_times__complex_Osimps_I2_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))) ).

% times_complex.simps(2)
tff(fact_9218_cmod__Im__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_real(re,Xb) = aa_complex_real(re,Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(im,Xb))),aa_real_real(abs_abs_real,aa_complex_real(im,Y))) ) ) ).

% cmod_Im_le_iff
tff(fact_9219_cmod__Re__le__iff,axiom,
    ! [Xb: complex,Y: complex] :
      ( ( aa_complex_real(im,Xb) = aa_complex_real(im,Y) )
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Xb)),aa_complex_real(real_V1022390504157884413omplex,Y))
      <=> aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_complex_real(re,Xb))),aa_real_real(abs_abs_real,aa_complex_real(re,Y))) ) ) ).

% cmod_Re_le_iff
tff(fact_9220_times__complex_Osimps_I1_J,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(times_times_complex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% times_complex.simps(1)
tff(fact_9221_plus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(plus_plus_complex(Xb),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% plus_complex.code
tff(fact_9222_scaleR__complex_Ocode,axiom,
    ! [R2: real,Xb: complex] : aa_complex_complex(real_V2046097035970521341omplex(R2),Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(re,Xb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_complex_real(im,Xb))) ).

% scaleR_complex.code
tff(fact_9223_minus__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(minus_minus_complex(Xb),Y) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y))) ).

% minus_complex.code
tff(fact_9224_csqrt__principal,axiom,
    ! [Zb: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(csqrt,Zb)))
      | ( ( aa_complex_real(re,aa_complex_complex(csqrt,Zb)) = 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,Zb))) ) ) ).

% csqrt_principal
tff(fact_9225_cmod__le,axiom,
    ! [Zb: complex] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(abs_abs_real,aa_complex_real(re,Zb))),aa_real_real(abs_abs_real,aa_complex_real(im,Zb)))) ).

% cmod_le
tff(fact_9226_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_9227_Re__exp,axiom,
    ! [Zb: complex] : aa_complex_real(re,aa_complex_complex(exp_complex,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Zb))),aa_real_real(cos_real,aa_complex_real(im,Zb))) ).

% Re_exp
tff(fact_9228_Im__exp,axiom,
    ! [Zb: complex] : aa_complex_real(im,aa_complex_complex(exp_complex,Zb)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(exp_real,aa_complex_real(re,Zb))),aa_real_real(sin_real,aa_complex_real(im,Zb))) ).

% Im_exp
tff(fact_9229_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_9230_times__complex_Ocode,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(times_times_complex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y)))) ).

% times_complex.code
tff(fact_9231_exp__eq__polar,axiom,
    ! [Zb: complex] : aa_complex_complex(exp_complex,Zb) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,aa_real_real(exp_real,aa_complex_real(re,Zb)))),aa_real_complex(cis,aa_complex_real(im,Zb))) ).

% exp_eq_polar
tff(fact_9232_cmod__power2,axiom,
    ! [Zb: complex] : aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),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,Zb)),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,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% cmod_power2
tff(fact_9233_Im__power2,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),aa_complex_real(re,Xb))),aa_complex_real(im,Xb)) ).

% Im_power2
tff(fact_9234_Re__power2,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_nat_complex(power_power_complex(Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

% Re_power2
tff(fact_9235_complex__eq__0,axiom,
    ! [Zb: complex] :
      ( ( Zb = 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,Zb)),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,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = zero_zero_real ) ) ).

% complex_eq_0
tff(fact_9236_norm__complex__def,axiom,
    ! [Zb: complex] : aa_complex_real(real_V1022390504157884413omplex,Zb) = 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,Zb)),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,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% norm_complex_def
tff(fact_9237_inverse__complex_Osimps_I1_J,axiom,
    ! [Xb: complex] : aa_complex_real(re,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(1)
tff(fact_9238_complex__neq__0,axiom,
    ! [Zb: complex] :
      ( ( Zb != 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,Zb)),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,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ) ).

% complex_neq_0
tff(fact_9239_Re__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Re_divide
tff(fact_9240_csqrt__unique,axiom,
    ! [W: complex,Zb: complex] :
      ( ( aa_nat_complex(power_power_complex(W),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) = Zb )
     => ( ( 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,Zb) = W ) ) ) ).

% csqrt_unique
tff(fact_9241_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,B2))
        | ( ( aa_complex_real(re,B2) = zero_zero_real )
          & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_complex_real(im,B2)) ) )
     => ( aa_complex_complex(csqrt,aa_nat_complex(power_power_complex(B2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = B2 ) ) ).

% csqrt_square
tff(fact_9242_inverse__complex_Osimps_I2_J,axiom,
    ! [Xb: complex] : aa_complex_real(im,aa_complex_complex(invers8013647133539491842omplex,Xb)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% inverse_complex.simps(2)
tff(fact_9243_Im__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% Im_divide
tff(fact_9244_complex__abs__le__norm,axiom,
    ! [Zb: 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,Zb))),aa_real_real(abs_abs_real,aa_complex_real(im,Zb)))),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,Zb))) ).

% complex_abs_le_norm
tff(fact_9245_complex__unit__circle,axiom,
    ! [Zb: complex] :
      ( ( Zb != 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,Zb)),aa_complex_real(real_V1022390504157884413omplex,Zb))),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,Zb)),aa_complex_real(real_V1022390504157884413omplex,Zb))),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = one_one_real ) ) ).

% complex_unit_circle
tff(fact_9246_inverse__complex_Ocode,axiom,
    ! [Xb: complex] : aa_complex_complex(invers8013647133539491842omplex,Xb) = aa_real_complex(complex2(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Xb)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(uminus_uminus_real,aa_complex_real(im,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Xb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% inverse_complex.code
tff(fact_9247_Complex__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(divide1717551699836669952omplex(Xb),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,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(im,Xb)),aa_complex_real(re,Y))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_complex_real(re,Xb)),aa_complex_real(im,Y)))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(re,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(im,Y)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% Complex_divide
tff(fact_9248_Im__Reals__divide,axiom,
    ! [R2: complex,Zb: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(R2),Zb)) = 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,Zb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Im_Reals_divide
tff(fact_9249_Re__Reals__divide,axiom,
    ! [R2: complex,Zb: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(R2),Zb)) = 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,Zb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ) ) ).

% Re_Reals_divide
tff(fact_9250_Re__divide__Reals,axiom,
    ! [R2: complex,Zb: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(Zb),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(re,Zb)),aa_complex_real(re,R2)) ) ) ).

% Re_divide_Reals
tff(fact_9251_real__eq__imaginary__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( Xb = aa_complex_complex(times_times_complex(imaginary_unit),Y) )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
tff(fact_9252_imaginary__eq__real__iff,axiom,
    ! [Y: complex,Xb: complex] :
      ( member_complex(Y,real_V2521375963428798218omplex)
     => ( member_complex(Xb,real_V2521375963428798218omplex)
       => ( ( aa_complex_complex(times_times_complex(imaginary_unit),Y) = Xb )
        <=> ( ( Xb = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
tff(fact_9253_Im__divide__Reals,axiom,
    ! [R2: complex,Zb: complex] :
      ( member_complex(R2,real_V2521375963428798218omplex)
     => ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(Zb),R2)) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_complex_real(im,Zb)),aa_complex_real(re,R2)) ) ) ).

% Im_divide_Reals
tff(fact_9254_complex__diff__cnj,axiom,
    ! [Zb: complex] : aa_complex_complex(minus_minus_complex(Zb),aa_complex_complex(cnj,Zb)) = 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,Zb)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_9255_complex__mult__cnj,axiom,
    ! [Zb: complex] : aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,Zb)) = 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,Zb)),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,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_mult_cnj
tff(fact_9256_integer__of__num_I3_J,axiom,
    ! [Nb: num] :
      code_integer_of_num(aa_num_num(bit1,Nb)) = $let(
        k: code_integer,
        k:= code_integer_of_num(Nb),
        plus_p5714425477246183910nteger(plus_p5714425477246183910nteger(k,k),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_9257_complex__cnj__mult,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(times_times_complex(Xb),Y)) = aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_mult
tff(fact_9258_complex__cnj__divide,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(divide1717551699836669952omplex(Xb),Y)) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_divide
tff(fact_9259_complex__cnj__add,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(plus_plus_complex(Xb),Y)) = aa_complex_complex(plus_plus_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_add
tff(fact_9260_complex__cnj__diff,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_complex(cnj,aa_complex_complex(minus_minus_complex(Xb),Y)) = aa_complex_complex(minus_minus_complex(aa_complex_complex(cnj,Xb)),aa_complex_complex(cnj,Y)) ).

% complex_cnj_diff
tff(fact_9261_complex__In__mult__cnj__zero,axiom,
    ! [Zb: complex] : aa_complex_real(im,aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,Zb))) = zero_zero_real ).

% complex_In_mult_cnj_zero
tff(fact_9262_Re__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = zero_zero_real )
    <=> ( aa_complex_real(re,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B2))) = zero_zero_real ) ) ).

% Re_complex_div_eq_0
tff(fact_9263_Im__complex__div__eq__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B2)) = zero_zero_real )
    <=> ( aa_complex_real(im,aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B2))) = zero_zero_real ) ) ).

% Im_complex_div_eq_0
tff(fact_9264_complex__mod__sqrt__Re__mult__cnj,axiom,
    ! [Zb: complex] : aa_complex_real(real_V1022390504157884413omplex,Zb) = aa_real_real(sqrt,aa_complex_real(re,aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,Zb)))) ).

% complex_mod_sqrt_Re_mult_cnj
tff(fact_9265_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_9266_Re__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B2))),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,B2)))),zero_zero_real) ) ).

% Re_complex_div_lt_0
tff(fact_9267_Re__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B2)))
    <=> 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,B2)))) ) ).

% Re_complex_div_gt_0
tff(fact_9268_Re__complex__div__ge__0,axiom,
    ! [A: complex,B2: 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),B2)))
    <=> 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,B2)))) ) ).

% Re_complex_div_ge_0
tff(fact_9269_Re__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B2))),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,B2)))),zero_zero_real) ) ).

% Re_complex_div_le_0
tff(fact_9270_Im__complex__div__lt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B2))),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,B2)))),zero_zero_real) ) ).

% Im_complex_div_lt_0
tff(fact_9271_Im__complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B2)))
    <=> 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,B2)))) ) ).

% Im_complex_div_gt_0
tff(fact_9272_Im__complex__div__ge__0,axiom,
    ! [A: complex,B2: 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),B2)))
    <=> 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,B2)))) ) ).

% Im_complex_div_ge_0
tff(fact_9273_Im__complex__div__le__0,axiom,
    ! [A: complex,B2: complex] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B2))),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,B2)))),zero_zero_real) ) ).

% Im_complex_div_le_0
tff(fact_9274_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_9275_complex__mod__mult__cnj,axiom,
    ! [Zb: complex] : aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,Zb))) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% complex_mod_mult_cnj
tff(fact_9276_complex__div__gt__0,axiom,
    ! [A: complex,B2: complex] :
      ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(re,aa_complex_complex(divide1717551699836669952omplex(A),B2)))
      <=> 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,B2)))) )
      & ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_complex_real(im,aa_complex_complex(divide1717551699836669952omplex(A),B2)))
      <=> 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,B2)))) ) ) ).

% complex_div_gt_0
tff(fact_9277_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_9278_complex__norm__square,axiom,
    ! [Zb: complex] : aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,Zb)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,Zb)) ).

% complex_norm_square
tff(fact_9279_complex__add__cnj,axiom,
    ! [Zb: complex] : aa_complex_complex(plus_plus_complex(Zb),aa_complex_complex(cnj,Zb)) = 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,Zb))) ).

% complex_add_cnj
tff(fact_9280_complex__div__cnj,axiom,
    ! [A: complex,B2: complex] : aa_complex_complex(divide1717551699836669952omplex(A),B2) = aa_complex_complex(divide1717551699836669952omplex(aa_complex_complex(times_times_complex(A),aa_complex_complex(cnj,B2))),aa_real_complex(real_V4546457046886955230omplex,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_complex_real(real_V1022390504157884413omplex,B2)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% complex_div_cnj
tff(fact_9281_cnj__add__mult__eq__Re,axiom,
    ! [Zb: complex,W: complex] : aa_complex_complex(plus_plus_complex(aa_complex_complex(times_times_complex(Zb),aa_complex_complex(cnj,W))),aa_complex_complex(times_times_complex(aa_complex_complex(cnj,Zb)),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(Zb),aa_complex_complex(cnj,W))))) ).

% cnj_add_mult_eq_Re
tff(fact_9282_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_9283_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_9284_card__lessThan,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_lessThan_nat,U)) = U ).

% card_lessThan
tff(fact_9285_card__Collect__less__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_ei,Nb))) = Nb ).

% card_Collect_less_nat
tff(fact_9286_card__atMost,axiom,
    ! [U: nat] : finite_card_nat(aa_nat_set_nat(set_ord_atMost_nat,U)) = aa_nat_nat(suc,U) ).

% card_atMost
tff(fact_9287_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_atLeastLessThan
tff(fact_9288_card__Collect__le__nat,axiom,
    ! [Nb: nat] : finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_eh,Nb))) = aa_nat_nat(suc,Nb) ).

% card_Collect_le_nat
tff(fact_9289_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or1269000886237332187st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(suc,U)),L) ).

% card_atLeastAtMost
tff(fact_9290_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or4662586982721622107an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_9291_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or1266510415728281911st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_9292_card__less__Suc2,axiom,
    ! [M5: set_nat,I: nat] :
      ( ~ member_nat(zero_zero_nat,M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fu(M5),I))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fv(M5),I))) ) ) ).

% card_less_Suc2
tff(fact_9293_card__less__Suc,axiom,
    ! [M5: set_nat,I: nat] :
      ( member_nat(zero_zero_nat,M5)
     => ( aa_nat_nat(suc,finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fu(M5),I)))) = finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fv(M5),I))) ) ) ).

% card_less_Suc
tff(fact_9294_card__less,axiom,
    ! [M5: set_nat,I: nat] :
      ( member_nat(zero_zero_nat,M5)
     => ( finite_card_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_fv(M5),I))) != zero_zero_nat ) ) ).

% card_less
tff(fact_9295_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_9296_subset__card__intvl__is__intvl,axiom,
    ! [A3: set_nat,K: nat] :
      ( ord_less_eq_set_nat(A3,aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A3))))
     => ( A3 = aa_nat_set_nat(set_or4665077453230672383an_nat(K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),finite_card_nat(A3))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_9297_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N: set_nat,Nb: nat] :
      ( ord_less_eq_set_nat(N,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(N)),Nb) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_9298_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_bb),aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),finite_card_nat(S2)))),aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bb),S2)) ).

% card_sum_le_nat_sum
tff(fact_9299_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_fj(C),Nb))) = Nb ) ) ) ).

% card_nth_roots
tff(fact_9300_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_er(Nb))) = Nb ) ) ).

% card_roots_unity_eq
tff(fact_9301_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_fw(K),code_divmod_abs(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% bit_cut_integer_code
tff(fact_9302_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_9303_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_fx(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_fy(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_9304_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),Xb)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz) )
                 => ( ( Y = some_nat(Ma) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_maxt_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_9305_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option_nat] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),Xb)
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = 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))) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw) )
               => ( ( Y = none_nat )
                 => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(none_P5556105721700978146at_nat,Uu,Uv,Uw)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz) )
                 => ( ( Y = some_nat(Mi) )
                   => ~ aa_VEBT_VEBT_o(accp_VEBT_VEBT(vEBT_vebt_mint_rel),vEBT_Node(some_P7363390416028606310at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mi),Ma)),Ux,Uy2,Uz)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_9306_prod__decode__aux_Osimps,axiom,
    ! [K: nat,Mb: nat] :
      nat_prod_decode_aux(K,Mb) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),K),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,K),Mb)),nat_prod_decode_aux(aa_nat_nat(suc,K),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,K)))) ).

% prod_decode_aux.simps
tff(fact_9307_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
    <=> aa_nat_o(case_nat_o($true,aTP_Lamm_fz),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_9308_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
    <=> aa_nat_o(case_nat_o($false,aTP_Lamm_ga),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_9309_less__eq__nat_Osimps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,Mb)),Nb)
    <=> aa_nat_o(case_nat_o($false,aa_nat_fun_nat_o(ord_less_eq_nat,Mb)),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_9310_max__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_gb,Nb)),Mb) ).

% max_Suc2
tff(fact_9311_max__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(aa_nat_nat(suc,Nb),aa_nat_fun_nat_nat(aTP_Lamm_gc,Nb)),Mb) ).

% max_Suc1
tff(fact_9312_diff__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)) ).

% diff_Suc
tff(fact_9313_prod__decode__aux_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) ) ) ).

% prod_decode_aux.elims
tff(fact_9314_pred__def,axiom,
    ! [Nat: nat] : aa_nat_nat(pred,Nat) = aa_nat_nat(case_nat_nat(zero_zero_nat,aTP_Lamm_bb),Nat) ).

% pred_def
tff(fact_9315_floor__real__def,axiom,
    ! [Xb: real] : aa_real_int(archim6058952711729229775r_real,Xb) = the_int(aTP_Lamm_gd(Xb)) ).

% floor_real_def
tff(fact_9316_floor__rat__def,axiom,
    ! [Xb: rat] : aa_rat_int(archim3151403230148437115or_rat,Xb) = the_int(aTP_Lamm_ge(Xb)) ).

% floor_rat_def
tff(fact_9317_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_9318_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)) = product_snd_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_mod_numeral
tff(fact_9319_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_9320_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_9321_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_9322_snd__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : product_snd_nat_nat(divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Mb),Nb) ).

% snd_divmod_nat
tff(fact_9323_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_9324_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_9325_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_9326_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_9327_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_9328_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
     => ~ ! [S3: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),S3)
           => ! [T: rat] :
                ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),T)
               => ( R2 != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,S3),T) ) ) ) ) ).

% obtain_pos_sum
tff(fact_9329_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,Xb),Y)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_9330_drop__bit__push__bit__int,axiom,
    ! [Mb: nat,Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Mb),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,Mb),Nb)),aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),K)) ).

% drop_bit_push_bit_int
tff(fact_9331_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa_int_int(bit_se8568078237143864401it_int(Nb),K) = aa_int_int(aa_int_fun_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_9332_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P2)) = produc4245557441103728435nt_int(aTP_Lamm_gf,quotient_of(P2)) ).

% rat_inverse_code
tff(fact_9333_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_9334_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)) = product_fst_nat_nat(unique5055182867167087721od_nat(one,K)) ).

% Suc_0_div_numeral
tff(fact_9335_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc6174133586879617921nteger(code_divmod_integer(K,L)) = modulo364778990260209775nteger(K,L) ).

% snd_divmod_integer
tff(fact_9336_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_9337_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) = zero_n2687167440665602831ol_nat(Nb = zero_zero_nat) ).

% drop_bit_of_Suc_0
tff(fact_9338_fst__divmod__nat,axiom,
    ! [Mb: nat,Nb: nat] : product_fst_nat_nat(divmod_nat(Mb,Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) ).

% fst_divmod_nat
tff(fact_9339_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_9340_diff__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(minus_minus_rat(Q2),R2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Q2),aa_rat_rat(uminus_uminus_rat,R2)) ).

% diff_rat_def
tff(fact_9341_divide__rat__def,axiom,
    ! [Q2: rat,R2: rat] : aa_rat_rat(divide_divide_rat(Q2),R2) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Q2),aa_rat_rat(inverse_inverse_rat,R2)) ).

% divide_rat_def
tff(fact_9342_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(divide_divide_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_gh(Q2),quotient_of(P2)) ).

% rat_divide_code
tff(fact_9343_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_gj(Q2),quotient_of(P2)) ).

% rat_times_code
tff(fact_9344_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_9345_quotient__of__div,axiom,
    ! [R2: rat,Nb: int,D: int] :
      ( ( quotient_of(R2) = product_Pair_int_int(Nb,D) )
     => ( R2 = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,Nb)),aa_int_rat(ring_1_of_int_rat,D)) ) ) ).

% quotient_of_div
tff(fact_9346_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_gl(Q2),quotient_of(P2)) ).

% rat_plus_code
tff(fact_9347_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa_rat_rat(minus_minus_rat(P2),Q2)) = produc4245557441103728435nt_int(aTP_Lamm_gn(Q2),quotient_of(P2)) ).

% rat_minus_code
tff(fact_9348_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_9349_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_9350_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( normalize(product_Pair_int_int(P2,Q2)) = normalize(product_Pair_int_int(R2,S)) )
         => ( aa_int_int(aa_int_fun_int_int(times_times_int,P2),S) = aa_int_int(aa_int_fun_int_int(times_times_int,R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_9351_drop__bit__nat__def,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,Nb),Mb) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)) ).

% drop_bit_nat_def
tff(fact_9352_one__mod__minus__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,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_9353_minus__one__mod__numeral,axiom,
    ! [Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),aa_num_int(numeral_numeral_int,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_9354_numeral__mod__minus__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = aa_int_int(uminus_uminus_int,aa_int_int(adjust_mod(aa_num_int(numeral_numeral_int,Nb)),product_snd_int_int(unique5052692396658037445od_int(Mb,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_9355_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_integer(K,L)) = divide6298287555418463151nteger(K,L) ).

% fst_divmod_integer
tff(fact_9356_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : produc8508995932063986495nteger(code_divmod_abs(K,L)) = divide6298287555418463151nteger(abs_abs_Code_integer(K),abs_abs_Code_integer(L)) ).

% fst_divmod_abs
tff(fact_9357_minus__numeral__mod__numeral,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),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(Mb,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_9358_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      aa_int_int(adjust_mod(L),R2) = $ite(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_9359_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y)))))) ).

% bezw.simps
tff(fact_9360_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_9361_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Y)
     => ( bezw(Xb,Y) = product_Pair_int_int(product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Y,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_9362_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_int_int] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,product_fst_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xaa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),Xaa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_9363_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(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a),aa_int_int(aa_int_fun_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(aa_int_fun_int_int(divide_divide_int,product_fst_int_int(P2)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,product_snd_int_int(P2)),a)) ) ) ) ).

% normalize_def
tff(fact_9364_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,aa_num_int(numeral_numeral_int,K))) = aa_rat_rat(divide_divide_rat(one_one_rat),aa_num_rat(numeral_numeral_rat,K)) ).

% Frct_code_post(5)
tff(fact_9365_gcd__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_gcd_int(Mb),Nb))
    <=> ( ( Mb != zero_zero_int )
        | ( Nb != zero_zero_int ) ) ) ).

% gcd_pos_int
tff(fact_9366_gcd__red__int,axiom,
    ! [Xb: int,Y: int] : aa_int_int(gcd_gcd_int(Xb),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) ).

% gcd_red_int
tff(fact_9367_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U3: int,V3: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,U3),Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,V3),Y)) = aa_int_int(gcd_gcd_int(Xb),Y) ).

% bezout_int
tff(fact_9368_gcd__mult__distrib__int,axiom,
    ! [K: int,Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,K)),aa_int_int(gcd_gcd_int(Mb),Nb)) = aa_int_int(gcd_gcd_int(aa_int_int(aa_int_fun_int_int(times_times_int,K),Mb)),aa_int_int(aa_int_fun_int_int(times_times_int,K),Nb)) ).

% gcd_mult_distrib_int
tff(fact_9369_gcd__le1__int,axiom,
    ! [A: int,B2: 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),B2)),A) ) ).

% gcd_le1_int
tff(fact_9370_gcd__le2__int,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(gcd_gcd_int(A),B2)),B2) ) ).

% gcd_le2_int
tff(fact_9371_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Y)
     => ( aa_int_int(gcd_gcd_int(Xb),Y) = aa_int_int(gcd_gcd_int(Y),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,Xb),Y)) ) ) ).

% gcd_non_0_int
tff(fact_9372_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(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(abs_abs_int,K)),aa_int_int(abs_abs_int,L))))) ).

% gcd_code_int
tff(fact_9373_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(product_Pair_int_int(aa_num_int(numeral_numeral_int,K),aa_num_int(numeral_numeral_int,L))) = aa_rat_rat(divide_divide_rat(aa_num_rat(numeral_numeral_rat,K)),aa_num_rat(numeral_numeral_rat,L)) ).

% Frct_code_post(6)
tff(fact_9374_prod__decode__aux_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod_nat_nat] :
      ( ( nat_prod_decode_aux(Xb,Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Xaa),Xb),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xaa),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),Xaa)),nat_prod_decode_aux(aa_nat_nat(suc,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xaa),aa_nat_nat(suc,Xb)))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(nat_pr5047031295181774490ux_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_9375_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( finite_finite_nat(S2)
     => ? [R3: fun_nat_nat] :
          ( strict1292158309912662752at_nat(R3,aa_nat_set_nat(set_ord_lessThan_nat,finite_card_nat(S2)))
          & ! [N5: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N5),finite_card_nat(S2))
             => member_nat(aa_nat_nat(R3,N5),S2) ) ) ) ).

% finite_enumerate
tff(fact_9376_gcd__1__nat,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_9377_gcd__Suc__0,axiom,
    ! [Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),aa_nat_nat(suc,zero_zero_nat)) = aa_nat_nat(suc,zero_zero_nat) ).

% gcd_Suc_0
tff(fact_9378_gcd__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb))
    <=> ( ( Mb != zero_zero_nat )
        | ( Nb != zero_zero_nat ) ) ) ).

% gcd_pos_nat
tff(fact_9379_gcd__red__nat,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) ).

% gcd_red_nat
tff(fact_9380_gcd__mult__distrib__nat,axiom,
    ! [K: nat,Mb: 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,Mb),Nb)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Mb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,K),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_9381_gcd__le2__nat,axiom,
    ! [B2: nat,A: nat] :
      ( ( B2 != 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),B2)),B2) ) ).

% gcd_le2_nat
tff(fact_9382_gcd__le1__nat,axiom,
    ! [A: nat,B2: 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),B2)),A) ) ).

% gcd_le1_nat
tff(fact_9383_gcd__diff2__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),Mb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff2_nat
tff(fact_9384_gcd__diff1__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),Mb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) ) ) ).

% gcd_diff1_nat
tff(fact_9385_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_9386_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y) = $ite(Y = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Y))) ).

% gcd_nat.simps
tff(fact_9387_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_9388_bezout__nat,axiom,
    ! [A: nat,B2: 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,B2),Y3)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2)) ) ).

% bezout_nat
tff(fact_9389_bezout__gcd__nat_H,axiom,
    ! [B2: 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,B2),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,B2),Y3)) = aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,A),B2) ) )
      | ( 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,B2),X3))
        & ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),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),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_9390_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_9391_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(bezw(Xb,Y))),aa_nat_int(semiri1314217659103216013at_int,Y))) ).

% bezw_aux
tff(fact_9392_nat__descend__induct,axiom,
    ! [Nb: nat,P: fun_nat_o,Mb: 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)
           => ( ! [I4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),I4)
                 => aa_nat_o(P,I4) )
             => aa_nat_o(P,K2) ) )
       => aa_nat_o(P,Mb) ) ) ).

% nat_descend_induct
tff(fact_9393_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Xaa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero_nat,Xb,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xaa),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Xb),Xaa))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(gcd_nat_rel),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_9394_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_go(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_9395_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or5832277885323065728an_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_9396_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),aa_num_int(neg_numeral_sub_int(Nb),one))) ).

% xor_minus_numerals(2)
tff(fact_9397_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(aa_num_int(neg_numeral_sub_int(Nb),one)),K)) ).

% xor_minus_numerals(1)
tff(fact_9398_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_finite_int(set_or5832277885323065728an_int(L,U)) ).

% finite_greaterThanLessThan_int
tff(fact_9399_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_9400_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or5832277885323065728an_int(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_9401_sub__BitM__One__eq,axiom,
    ! [Nb: num] : aa_num_int(neg_numeral_sub_int(aa_num_num(bitM,Nb)),one) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_int(neg_numeral_sub_int(Nb),one)) ).

% sub_BitM_One_eq
tff(fact_9402_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_finite_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) ).

% finite_greaterThanLessThan
tff(fact_9403_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),aa_nat_nat(suc,L)) ).

% card_greaterThanLessThan
tff(fact_9404_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_9405_tanh__real__bounds,axiom,
    ! [Xb: real] : member_real(aa_real_real(tanh_real,Xb),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) ).

% tanh_real_bounds
tff(fact_9406_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1623282765462674594er_nat(ord_max_nat,zero_zero_nat,aTP_Lamm_eh,aTP_Lamm_ei) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_9407_Suc__funpow,axiom,
    ! [Nb: nat] : compow_fun_nat_nat(Nb,suc) = aa_nat_fun_nat_nat(plus_plus_nat,Nb) ).

% Suc_funpow
tff(fact_9408_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_gp,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ) ).

% int_of_integer_code
tff(fact_9409_plus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(plus_p5714425477246183910nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_9410_times__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Xb),Xaa)) = aa_int_int(aa_int_fun_int_int(times_times_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_9411_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(minus_8373710615458151222nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_9412_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(divide6298287555418463151nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% divide_integer.rep_eq
tff(fact_9413_modulo__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(modulo364778990260209775nteger(Xb,Xaa)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% modulo_integer.rep_eq
tff(fact_9414_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,Xb),Xaa)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_9415_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,K),L)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_9416_times__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_gr),Xaa),Xb)) ).

% times_int.abs_eq
tff(fact_9417_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( finite_finite_nat(M5)
     => ( gcd_Gcd_nat(M5) = gcd_Gcd_nat(minus_minus_set_nat(M5,insert_nat(zero_zero_nat,bot_bot_set_nat))) ) ) ).

% Gcd_remove0_nat
tff(fact_9418_Gcd__nat__eq__one,axiom,
    ! [N: set_nat] :
      ( member_nat(one_one_nat,N)
     => ( gcd_Gcd_nat(N) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_9419_nat_Oabs__eq,axiom,
    ! [Xb: product_prod_nat_nat] : aa_int_nat(nat2,aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),Xb) ).

% nat.abs_eq
tff(fact_9420_one__int__def,axiom,
    one_one_int = aa_Pro4240683299535347697at_int(abs_Integ,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int_def
tff(fact_9421_less__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_gt),Xaa),Xb) ) ).

% less_int.abs_eq
tff(fact_9422_less__eq__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb))
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_gv),Xaa),Xb) ) ).

% less_eq_int.abs_eq
tff(fact_9423_plus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_gx),Xaa),Xb)) ).

% plus_int.abs_eq
tff(fact_9424_minus__int_Oabs__eq,axiom,
    ! [Xaa: product_prod_nat_nat,Xb: product_prod_nat_nat] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro4240683299535347697at_int(abs_Integ,Xaa)),aa_Pro4240683299535347697at_int(abs_Integ,Xb)) = aa_Pro4240683299535347697at_int(abs_Integ,aa_Pro262675715343573244at_nat(aa_Pro8023792847356852296at_nat(produc4528391901023682385at_nat(aTP_Lamm_gz),Xaa),Xb)) ).

% minus_int.abs_eq
tff(fact_9425_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_9426_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_9427_num__of__nat_Osimps_I1_J,axiom,
    aa_nat_num(num_of_nat,zero_zero_nat) = one ).

% num_of_nat.simps(1)
tff(fact_9428_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_9429_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_9430_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_9431_num__of__nat__plus__distrib,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) = aa_num_num(plus_plus_num(aa_nat_num(num_of_nat,Mb)),aa_nat_num(num_of_nat,Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_9432_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_9433_less__eq__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_gv),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_9434_less__int_Orep__eq,axiom,
    ! [Xb: int,Xaa: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,Xb),Xaa)
    <=> aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(produc6395135316500328296_nat_o(aTP_Lamm_gt),aa_int1417691210696471063at_nat(rep_Integ,Xb)),aa_int1417691210696471063at_nat(rep_Integ,Xaa)) ) ).

% less_int.rep_eq
tff(fact_9435_nat_Orep__eq,axiom,
    ! [Xb: int] : aa_int_nat(nat2,Xb) = aa_Pro4243173770044397973at_nat(produc6842872674320459806at_nat(minus_minus_nat),aa_int1417691210696471063at_nat(rep_Integ,Xb)) ).

% nat.rep_eq
tff(fact_9436_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Nb: nat,J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J))),Nb) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Nb)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_9437_prod__encode__def,axiom,
    nat_prod_encode = produc6842872674320459806at_nat(aTP_Lamm_ha) ).

% prod_encode_def
tff(fact_9438_pow_Osimps_I3_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(pow(Xb),aa_num_num(bit1,Y)) = aa_num_num(times_times_num(aa_num_num(sqr,aa_num_num(pow(Xb),Y))),Xb) ).

% pow.simps(3)
tff(fact_9439_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_finite_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) ).

% finite_greaterThanAtMost
tff(fact_9440_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : finite_card_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(L),U)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,U),L) ).

% card_greaterThanAtMost
tff(fact_9441_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_9442_sqr_Osimps_I1_J,axiom,
    aa_num_num(sqr,one) = one ).

% sqr.simps(1)
tff(fact_9443_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_9444_sqr__conv__mult,axiom,
    ! [Xb: num] : aa_num_num(sqr,Xb) = aa_num_num(times_times_num(Xb),Xb) ).

% sqr_conv_mult
tff(fact_9445_le__prod__encode__1,axiom,
    ! [A: nat,B2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B2))) ).

% le_prod_encode_1
tff(fact_9446_le__prod__encode__2,axiom,
    ! [B2: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B2),aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),B2))) ).

% le_prod_encode_2
tff(fact_9447_pow_Osimps_I2_J,axiom,
    ! [Xb: num,Y: num] : aa_num_num(pow(Xb),aa_num_num(bit0,Y)) = aa_num_num(sqr,aa_num_num(pow(Xb),Y)) ).

% pow.simps(2)
tff(fact_9448_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(plus_plus_num(aa_num_num(sqr,Nb)),Nb))) ).

% sqr.simps(3)
tff(fact_9449_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,Mb: nat] : aa_Pro4243173770044397973at_nat(nat_prod_encode,nat_prod_decode_aux(K,Mb)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb) ).

% prod_encode_prod_decode_aux
tff(fact_9450_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( (Y)
          <=> ( Xaa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_9451_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( Xaa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n: nat,
                      n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                      $let(
                        m2: nat,
                        m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                        ( ! [X: vEBT_VEBT] :
                            ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                           => aa_nat_o(vEBT_VEBT_valid(X),n) )
                        & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                        & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                        & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                            & ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                               => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_9452_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( ( ? [Uu: $o,Uv: $o] : Xb = vEBT_Leaf((Uu),(Uv))
         => ( Xaa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n: nat,
                    n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                    $let(
                      m2: nat,
                      m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                      ( ! [X: vEBT_VEBT] :
                          ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                         => aa_nat_o(vEBT_VEBT_valid(X),n) )
                      & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                      & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                      & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                          & ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                             => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_9453_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_finite_int(set_or6656581121297822940st_int(L,U)) ).

% finite_greaterThanAtMost_int
tff(fact_9454_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(set_or6656581121297822940st_int(L,U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_9455_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int),U) = set_or6656581121297822940st_int(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_9456_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),
              ( ! [X: vEBT_VEBT] :
                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
                 => aa_nat_o(vEBT_VEBT_valid(X),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) )
              & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summarya),X6)
                  & ! [X: vEBT_VEBT] :
                      ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeLista))
                     => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Dega),TreeLista),Summarya),n),m2)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_9457_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
      <=> (Y) )
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one_nat ) )
               => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa)) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) )
                 => ~ accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_9458_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n: nat,
                          n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                          $let(
                            m2: nat,
                            m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                            ( ! [X: vEBT_VEBT] :
                                ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                               => aa_nat_o(vEBT_VEBT_valid(X),n) )
                            & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                            & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                            & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                   => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_9459_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa_nat_o(vEBT_VEBT_valid(Xb),Xaa)
     => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(Xb,Xaa))
       => ( ! [Uu: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu),(Uv)) )
             => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Leaf((Uu),(Uv)),Xaa))
               => ( Xaa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList,Summary) )
               => ( accp_P2887432264394892906BT_nat(vEBT_VEBT_valid_rel,produc738532404422230701BT_nat(vEBT_Node(Mima,Deg,TreeList,Summary),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n: nat,
                        n:= aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),
                        $let(
                          m2: nat,
                          m2:= aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Deg),n),
                          ( ! [X: vEBT_VEBT] :
                              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                             => aa_nat_o(vEBT_VEBT_valid(X),n) )
                          & aa_nat_o(vEBT_VEBT_valid(Summary),m2)
                          & ( size_s6755466524823107622T_VEBT(TreeList) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),m2) )
                          & case_o184042715313410164at_nat(( ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member_VEBT_VEBT(X,set_VEBT_VEBT2(TreeList))
                                 => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ) ),produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Deg),TreeList),Summary),n),m2)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_9460_GreatestI__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: 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),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_nat
tff(fact_9461_Greatest__le__nat,axiom,
    ! [P: fun_nat_o,K: nat,B2: 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),B2) )
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),order_Greatest_nat(P)) ) ) ).

% Greatest_le_nat
tff(fact_9462_GreatestI__ex__nat,axiom,
    ! [P: fun_nat_o,B2: nat] :
      ( ? [X_12: nat] : aa_nat_o(P,X_12)
     => ( ! [Y3: nat] :
            ( aa_nat_o(P,Y3)
           => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y3),B2) )
       => aa_nat_o(P,order_Greatest_nat(P)) ) ) ).

% GreatestI_ex_nat
tff(fact_9463_rat__floor__lemma,axiom,
    ! [A: int,B2: int] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2))),aa_int_rat(fract(A),B2))
      & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_9464_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Xb: nat,Y: nat] :
      image_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hc,C),aa_nat_set_nat(set_or4665077453230672383an_nat(Xb),Y)) = $ite(
        aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),Y),
        aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Xb),C)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Y),C)),
        $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y),insert_nat(zero_zero_nat,bot_bot_set_nat),bot_bot_set_nat) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_9465_take__bit__numeral__minus__numeral__int,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,aTP_Lamm_hd(Mb),bit_take_bit_num(aa_num_nat(numeral_numeral_nat,Mb),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_9466_take__bit__num__simps_I1_J,axiom,
    ! [Mb: num] : bit_take_bit_num(zero_zero_nat,Mb) = none_num ).

% take_bit_num_simps(1)
tff(fact_9467_bij__betw__Suc,axiom,
    ! [M5: set_nat,N: set_nat] :
      ( bij_betw_nat_nat(suc,M5,N)
    <=> ( image_nat_nat(suc,M5) = N ) ) ).

% bij_betw_Suc
tff(fact_9468_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_9469_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_9470_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or1269000886237332187st_nat(I),J)) = aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_9471_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : image_nat_nat(suc,aa_nat_set_nat(set_or4665077453230672383an_nat(I),J)) = aa_nat_set_nat(set_or4665077453230672383an_nat(aa_nat_nat(suc,I)),aa_nat_nat(suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_9472_mult__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ).

% mult_rat
tff(fact_9473_divide__rat,axiom,
    ! [A: int,B2: int,C: int,D: int] : aa_rat_rat(divide_divide_rat(aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),C)) ).

% divide_rat
tff(fact_9474_floor__Fract,axiom,
    ! [A: int,B2: int] : aa_rat_int(archim3151403230148437115or_rat,aa_int_rat(fract(A),B2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B2) ).

% floor_Fract
tff(fact_9475_less__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ).

% less_rat
tff(fact_9476_add__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ).

% add_rat
tff(fact_9477_le__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D))
        <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D))) ) ) ) ).

% le_rat
tff(fact_9478_diff__rat,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( aa_rat_rat(minus_minus_rat(aa_int_rat(fract(A),B2)),aa_int_rat(fract(C),D)) = aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2))),aa_int_int(aa_int_fun_int_int(times_times_int,B2),D)) ) ) ) ).

% diff_rat
tff(fact_9479_sgn__rat,axiom,
    ! [A: int,B2: int] : aa_rat_rat(sgn_sgn_rat,aa_int_rat(fract(A),B2)) = aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B2))) ).

% sgn_rat
tff(fact_9480_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_he,Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_9481_zero__notin__Suc__image,axiom,
    ! [A3: set_nat] : ~ member_nat(zero_zero_nat,image_nat_nat(suc,A3)) ).

% zero_notin_Suc_image
tff(fact_9482_Rat__induct__pos,axiom,
    ! [P: fun_rat_o,Q2: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => aa_rat_o(P,aa_int_rat(fract(A4),B4)) )
     => aa_rat_o(P,Q2) ) ).

% Rat_induct_pos
tff(fact_9483_eq__rat_I1_J,axiom,
    ! [B2: int,D: int,A: int,C: int] :
      ( ( B2 != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( aa_int_rat(fract(A),B2) = aa_int_rat(fract(C),D) )
        <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),D) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B2) ) ) ) ) ).

% eq_rat(1)
tff(fact_9484_mult__rat__cancel,axiom,
    ! [C: int,A: int,B2: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B2)) = aa_int_rat(fract(A),B2) ) ) ).

% mult_rat_cancel
tff(fact_9485_Fract__coprime,axiom,
    ! [A: int,B2: int] : aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(gcd_gcd_int(A),B2))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),aa_int_int(gcd_gcd_int(A),B2))) = aa_int_rat(fract(A),B2) ).

% Fract_coprime
tff(fact_9486_Fract__of__int__quotient,axiom,
    ! [K: int,L: int] : aa_int_rat(fract(K),L) = aa_rat_rat(divide_divide_rat(aa_int_rat(ring_1_of_int_rat,K)),aa_int_rat(ring_1_of_int_rat,L)) ).

% Fract_of_int_quotient
tff(fact_9487_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_9488_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_9489_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_9490_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_9491_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_9492_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_9493_Fract__less__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ) ).

% Fract_less_zero_iff
tff(fact_9494_zero__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ) ).

% zero_less_Fract_iff
tff(fact_9495_Fract__less__one__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_int_rat(fract(A),B2)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B2) ) ) ).

% Fract_less_one_iff
tff(fact_9496_one__less__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,one_one_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B2),A) ) ) ).

% one_less_Fract_iff
tff(fact_9497_Fract__add__one,axiom,
    ! [Nb: int,Mb: int] :
      ( ( Nb != zero_zero_int )
     => ( aa_int_rat(fract(aa_int_int(aa_int_fun_int_int(plus_plus_int,Mb),Nb)),Nb) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(fract(Mb),Nb)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_9498_Fract__le__zero__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),zero_zero_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ) ).

% Fract_le_zero_iff
tff(fact_9499_zero__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,zero_zero_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ) ).

% zero_le_Fract_iff
tff(fact_9500_one__le__Fract__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,one_one_rat),aa_int_rat(fract(A),B2))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B2),A) ) ) ).

% one_le_Fract_iff
tff(fact_9501_Fract__le__one__iff,axiom,
    ! [B2: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B2)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(fract(A),B2)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B2) ) ) ).

% Fract_le_one_iff
tff(fact_9502_take__bit__num__def,axiom,
    ! [Nb: nat,Mb: num] :
      bit_take_bit_num(Nb,Mb) = $ite(aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb)) = zero_zero_nat,none_num,some_num(aa_nat_num(num_of_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,Nb),aa_num_nat(numeral_numeral_nat,Mb))))) ).

% take_bit_num_def
tff(fact_9503_and__minus__numerals_I3_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(3)
tff(fact_9504_and__minus__numerals_I7_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bitM,Nb))) ).

% and_minus_numerals(7)
tff(fact_9505_and__minus__numerals_I4_J,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(4)
tff(fact_9506_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Nb,Mb))) ).

% take_bit_num_simps(4)
tff(fact_9507_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(aa_nat_nat(suc,Nb),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_hf,bit_take_bit_num(Nb,Mb)) ).

% take_bit_num_simps(3)
tff(fact_9508_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit1,Mb)) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb))) ).

% take_bit_num_simps(7)
tff(fact_9509_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,Mb: num] : bit_take_bit_num(aa_num_nat(numeral_numeral_nat,R2),aa_num_num(bit0,Mb)) = case_o6005452278849405969um_num(none_num,aTP_Lamm_hf,bit_take_bit_num(aa_num_nat(pred_numeral,R2),Mb)) ).

% take_bit_num_simps(6)
tff(fact_9510_and__minus__numerals_I8_J,axiom,
    ! [Nb: num,Mb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,aa_num_num(bit1,Nb)))),aa_num_int(numeral_numeral_int,Mb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,aa_num_num(bit0,Nb))) ).

% and_minus_numerals(8)
tff(fact_9511_and__not__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit0,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(8)
tff(fact_9512_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one,one) = none_num ).

% and_not_num.simps(1)
tff(fact_9513_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_9514_and__not__num_Osimps_I4_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit0,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(4)
tff(fact_9515_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_9516_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit0,Mb)) = case_nat_option_num(none_num,aTP_Lamm_hh(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_9517_image__int__atLeastAtMost,axiom,
    ! [A: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or1269000886237332187st_nat(A),B2)) = set_or1266510415728281911st_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastAtMost
tff(fact_9518_image__int__atLeastLessThan,axiom,
    ! [A: nat,B2: nat] : image_nat_int(semiri1314217659103216013at_int,aa_nat_set_nat(set_or4665077453230672383an_nat(A),B2)) = set_or4662586982721622107an_int(aa_nat_int(semiri1314217659103216013at_int,A),aa_nat_int(semiri1314217659103216013at_int,B2)) ).

% image_int_atLeastLessThan
tff(fact_9519_and__not__num_Osimps_I7_J,axiom,
    ! [Mb: num] : bit_and_not_num(aa_num_num(bit1,Mb),one) = some_num(aa_num_num(bit0,Mb)) ).

% and_not_num.simps(7)
tff(fact_9520_and__not__num__eq__Some__iff,axiom,
    ! [Mb: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(Mb,Nb) = some_num(Q2) )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = aa_num_int(numeral_numeral_int,Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_9521_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,aa_num_num(bit1,Mb)) = case_nat_option_num(none_num,aTP_Lamm_hi(Mb),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_9522_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : image_int_int(aTP_Lamm_hj(L),set_or4662586982721622107an_int(zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),L))) = set_or4662586982721622107an_int(L,U) ).

% image_add_int_atLeastLessThan
tff(fact_9523_and__not__num__eq__None__iff,axiom,
    ! [Mb: num,Nb: num] :
      ( ( bit_and_not_num(Mb,Nb) = none_num )
    <=> ( aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
tff(fact_9524_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_9525_int__numeral__and__not__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_num_int(numeral_numeral_int,Mb)),aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Nb))) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Mb,Nb)) ).

% int_numeral_and_not_num
tff(fact_9526_int__numeral__not__and__num,axiom,
    ! [Mb: num,Nb: num] : aa_int_int(bit_se725231765392027082nd_int(aa_int_int(bit_ri7919022796975470100ot_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)) = case_option_int_num(zero_zero_int,numeral_numeral_int,bit_and_not_num(Nb,Mb)) ).

% int_numeral_not_and_num
tff(fact_9527_positive__rat,axiom,
    ! [A: int,B2: int] :
      ( aa_rat_o(positive,aa_int_rat(fract(A),B2))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,A),B2)) ) ).

% positive_rat
tff(fact_9528_Rat_Opositive__add,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(positive,Xb)
     => ( aa_rat_o(positive,Y)
       => aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Xb),Y)) ) ) ).

% Rat.positive_add
tff(fact_9529_Rat_Opositive__mult,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(positive,Xb)
     => ( aa_rat_o(positive,Y)
       => aa_rat_o(positive,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Xb),Y)) ) ) ).

% Rat.positive_mult
tff(fact_9530_less__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Xb),Y)
    <=> aa_rat_o(positive,aa_rat_rat(minus_minus_rat(Y),Xb)) ) ).

% less_rat_def
tff(fact_9531_Rat_Opositive_Orep__eq,axiom,
    ! [Xb: rat] :
      ( aa_rat_o(positive,Xb)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xb))),product_snd_int_int(aa_rat7754553326175492643nt_int(rep_Rat,Xb)))) ) ).

% Rat.positive.rep_eq
tff(fact_9532_and__not__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xb = one )
           => ( ? [N2: num] : Xaa = aa_num_num(bit0,N2)
             => ( Y != some_num(one) ) ) )
         => ( ( ( Xb = one )
             => ( ? [N2: num] : Xaa = aa_num_num(bit1,N2)
               => ( Y != none_num ) ) )
           => ( ! [M3: num] :
                  ( ( Xb = aa_num_num(bit0,M3) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) )
                 => ( ! [M3: num] :
                        ( ( Xb = aa_num_num(bit1,M3) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,bit_and_not_num(M3,N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,bit_and_not_num(M3,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_9533_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,Mb: num] : bit_take_bit_num(Nb,Mb) = produc478579273971653890on_num(aTP_Lamm_hn,product_Pair_nat_num(Nb,Mb)) ).

% Bit_Operations.take_bit_num_code
tff(fact_9534_and__not__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(5)
tff(fact_9535_and__not__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit1,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(9)
tff(fact_9536_and__not__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : bit_and_not_num(aa_num_num(bit0,Mb),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,bit_and_not_num(Mb,Nb)) ).

% and_not_num.simps(6)
tff(fact_9537_and__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != some_num(one) ) ) )
       => ( ( ( Xb = one )
           => ( ? [N2: num] : Xaa = aa_num_num(bit0,N2)
             => ( Y != none_num ) ) )
         => ( ( ( Xb = one )
             => ( ? [N2: num] : Xaa = aa_num_num(bit1,N2)
               => ( Y != some_num(one) ) ) )
           => ( ( ? [M3: num] : Xb = aa_num_num(bit0,M3)
               => ( ( Xaa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
                 => ( ( ? [M3: num] : Xb = aa_num_num(bit1,M3)
                     => ( ( Xaa = one )
                       => ( Y != some_num(one) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) )
                     => ~ ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Y != case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_9538_xor__num_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Xb),Xaa) = Y )
     => ( ( ( Xb = one )
         => ( ( Xaa = one )
           => ( Y != none_num ) ) )
       => ( ( ( Xb = one )
           => ! [N2: num] :
                ( ( Xaa = aa_num_num(bit0,N2) )
               => ( Y != some_num(aa_num_num(bit1,N2)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit1,N2) )
                 => ( Y != some_num(aa_num_num(bit0,N2)) ) ) )
           => ( ! [M3: num] :
                  ( ( Xb = aa_num_num(bit0,M3) )
                 => ( ( Xaa = one )
                   => ( Y != some_num(aa_num_num(bit1,M3)) ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ! [N2: num] :
                        ( ( Xaa = aa_num_num(bit0,N2) )
                       => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit1,N2) )
                         => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( Xb = aa_num_num(bit1,M3) )
                       => ( ( Xaa = one )
                         => ( Y != some_num(aa_num_num(bit0,M3)) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ! [N2: num] :
                              ( ( Xaa = aa_num_num(bit0,N2) )
                             => ( Y != some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) ) ) )
                     => ~ ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit1,N2) )
                               => ( Y != map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_9539_xor__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(8)
tff(fact_9540_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_9541_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_9542_and__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(5)
tff(fact_9543_xor__num_Osimps_I5_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(5)
tff(fact_9544_and__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),one) = some_num(one) ).

% and_num.simps(7)
tff(fact_9545_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_9546_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_9547_and__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),one) = none_num ).

% and_num.simps(4)
tff(fact_9548_xor__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb)) ).

% xor_num.simps(9)
tff(fact_9549_and__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(6)
tff(fact_9550_and__num_Osimps_I8_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit0,Nb)) = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(8)
tff(fact_9551_xor__num_Osimps_I7_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit1,Mb)),one) = some_num(aa_num_num(bit0,Mb)) ).

% xor_num.simps(7)
tff(fact_9552_xor__num_Osimps_I4_J,axiom,
    ! [Mb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),one) = some_num(aa_num_num(bit1,Mb)) ).

% xor_num.simps(4)
tff(fact_9553_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_9554_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_9555_and__num_Osimps_I9_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,aa_num_num(bit1,Mb)),aa_num_num(bit1,Nb)) = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Mb),Nb)) ).

% and_num.simps(9)
tff(fact_9556_xor__num_Osimps_I6_J,axiom,
    ! [Mb: num,Nb: num] : aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,aa_num_num(bit0,Mb)),aa_num_num(bit1,Nb)) = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Mb),Nb))) ).

% xor_num.simps(6)
tff(fact_9557_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_9558_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_9559_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_ho,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% num_of_integer_code
tff(fact_9560_min__Suc__Suc,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Mb)),aa_nat_nat(suc,Nb)) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_Suc_Suc
tff(fact_9561_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_9562_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_9563_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_9564_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_9565_nat__mult__min__right,axiom,
    ! [Mb: nat,Nb: nat,Q2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Nb),Q2)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)) ).

% nat_mult_min_right
tff(fact_9566_nat__mult__min__left,axiom,
    ! [Mb: 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,Mb),Nb)),Q2) = aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Q2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Nb),Q2)) ).

% nat_mult_min_left
tff(fact_9567_min__diff,axiom,
    ! [Mb: nat,I: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),I)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),I)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)),I) ).

% min_diff
tff(fact_9568_concat__bit__assoc__sym,axiom,
    ! [Mb: nat,Nb: nat,K: int,L: int,R2: int] : aa_int_int(bit_concat_bit(Mb,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,Mb),Nb),K),aa_int_int(bit_concat_bit(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_9569_take__bit__concat__bit__eq,axiom,
    ! [Mb: nat,Nb: nat,K: int,L: int] : aa_int_int(bit_se2923211474154528505it_int(Mb),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,Mb),Nb),K),aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Mb),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_9570_min__Suc1,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,aa_nat_nat(suc,Nb)),Mb) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_hp,Nb)),Mb) ).

% min_Suc1
tff(fact_9571_min__Suc2,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),aa_nat_nat(suc,Nb)) = aa_nat_nat(case_nat_nat(zero_zero_nat,aa_nat_fun_nat_nat(aTP_Lamm_hq,Nb)),Mb) ).

% min_Suc2
tff(fact_9572_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( finite_finite_nat(S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,finite_card_nat(S2)),aa_nat_nat(suc,lattic8265883725875713057ax_nat(S2))) ) ).

% card_le_Suc_Max
tff(fact_9573_divide__nat__def,axiom,
    ! [Mb: nat,Nb: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Mb),Nb) = $ite(Nb = zero_zero_nat,zero_zero_nat,lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hr(Mb),Nb)))) ).

% divide_nat_def
tff(fact_9574_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,Mb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb) = lattic8265883725875713057ax_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_hs(Nb),Mb))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_9575_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_9576_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_9577_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_9578_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I),J)) = cons_nat(aa_nat_nat(suc,I),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_9579_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list_int] :
      upto_aux(I,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I),Js,upto_aux(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_9580_list__encode_Osimps_I2_J,axiom,
    ! [Xb: nat,Xs: list_nat] : nat_list_encode(cons_nat(Xb,Xs)) = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_9581_list__encode_Oelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( ( ( Xb = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs2: list_nat] :
              ( ( Xb = cons_nat(X3,Xs2) )
             => ( Y != aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_9582_quotient__of__def,axiom,
    ! [Xb: rat] : quotient_of(Xb) = the_Pr4378521158711661632nt_int(aTP_Lamm_ht(Xb)) ).

% quotient_of_def
tff(fact_9583_coprime__abs__right__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_int_int(abs_abs_int,L))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_right_iff
tff(fact_9584_coprime__abs__left__iff,axiom,
    ! [K: int,L: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_int_int(abs_abs_int,K)),L)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),L) ) ).

% coprime_abs_left_iff
tff(fact_9585_normalize__stable,axiom,
    ! [Q2: int,P2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Q2)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,P2),Q2)
       => ( normalize(product_Pair_int_int(P2,Q2)) = product_Pair_int_int(P2,Q2) ) ) ) ).

% normalize_stable
tff(fact_9586_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),D)
     => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,B2),C)
       => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,B2)),aa_int_int(abs_abs_int,D)) )
        <=> ( ( aa_int_int(abs_abs_int,A) = aa_int_int(abs_abs_int,B2) )
            & ( aa_int_int(abs_abs_int,C) = aa_int_int(abs_abs_int,D) ) ) ) ) ) ).

% coprime_crossproduct_int
tff(fact_9587_Rat__cases,axiom,
    ! [Q2: rat] :
      ~ ! [A4: int,B4: int] :
          ( ( Q2 = aa_int_rat(fract(A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) ).

% Rat_cases
tff(fact_9588_Rat__induct,axiom,
    ! [P: fun_rat_o,Q2: rat] :
      ( ! [A4: int,B4: int] :
          ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
         => ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4)
           => aa_rat_o(P,aa_int_rat(fract(A4),B4)) ) )
     => aa_rat_o(P,Q2) ) ).

% Rat_induct
tff(fact_9589_coprime__common__divisor__int,axiom,
    ! [A: int,B2: int,Xb: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),B2)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Xb),B2)
         => ( aa_int_int(abs_abs_int,Xb) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_9590_Rat__cases__nonzero,axiom,
    ! [Q2: rat] :
      ( ! [A4: int,B4: int] :
          ( ( Q2 = aa_int_rat(fract(A4),B4) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B4)
           => ( ( A4 != zero_zero_int )
             => ~ aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A4),B4) ) ) )
     => ( Q2 = zero_zero_rat ) ) ).

% Rat_cases_nonzero
tff(fact_9591_card__length__sum__list__rec,axiom,
    ! [Mb: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),Mb)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(Mb),N))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hv(Mb),N)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hw(Mb),N)))) ) ) ).

% card_length_sum_list_rec
tff(fact_9592_card__length__sum__list,axiom,
    ! [Mb: nat,N: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(Mb),N))) = 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,N),Mb)),one_one_nat)),N) ).

% card_length_sum_list
tff(fact_9593_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod_int_int] :
      ( ( R2 = aa_int_rat(fract(product_fst_int_int(X3)),product_snd_int_int(X3)) )
      & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(X3))
      & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(X3)),product_snd_int_int(X3))
      & ! [Y4: product_prod_int_int] :
          ( ( ( R2 = aa_int_rat(fract(product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
            & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Y4))
            & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Y4)),product_snd_int_int(Y4)) )
         => ( Y4 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_9594_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_9595_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_9596_coprime__int__iff,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Mb)),aa_nat_int(semiri1314217659103216013at_int,Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Mb),Nb) ) ).

% coprime_int_iff
tff(fact_9597_coprime__nat__abs__right__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_int_nat(nat2,aa_int_int(abs_abs_int,K)))
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,aa_nat_int(semiri1314217659103216013at_int,Nb)),K) ) ).

% coprime_nat_abs_right_iff
tff(fact_9598_coprime__nat__abs__left__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_int_nat(nat2,aa_int_int(abs_abs_int,K))),Nb)
    <=> aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,K),aa_nat_int(semiri1314217659103216013at_int,Nb)) ) ).

% coprime_nat_abs_left_iff
tff(fact_9599_coprime__common__divisor__nat,axiom,
    ! [A: nat,B2: nat,Xb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Xb),B2)
         => ( Xb = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_9600_coprime__Suc__0__right,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,zero_zero_nat)) ).

% coprime_Suc_0_right
tff(fact_9601_coprime__Suc__0__left,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,zero_zero_nat)),Nb) ).

% coprime_Suc_0_left
tff(fact_9602_coprime__Suc__left__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(suc,Nb)),Nb) ).

% coprime_Suc_left_nat
tff(fact_9603_coprime__Suc__right__nat,axiom,
    ! [Nb: nat] : aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(suc,Nb)) ).

% coprime_Suc_right_nat
tff(fact_9604_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),D)
     => ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,B2),C)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B2),D) )
        <=> ( ( A = B2 )
            & ( C = D ) ) ) ) ) ).

% coprime_crossproduct_nat
tff(fact_9605_coprime__diff__one__right__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,Nb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_9606_coprime__diff__one__left__nat,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)),Nb) ) ).

% coprime_diff_one_left_nat
tff(fact_9607_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(I,J))
     => ( upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_9608_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil
tff(fact_9609_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int = upto(I,J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I) ) ).

% upto_Nil2
tff(fact_9610_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I)
     => ( upto(I,J) = nil_int ) ) ).

% upto_empty
tff(fact_9611_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(upto(I,J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_9612_length__upto,axiom,
    ! [I: int,J: int] : size_size_list_int(upto(I,J)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),I)),one_one_int)) ).

% length_upto
tff(fact_9613_upto__rec__numeral_I1_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_9614_upto__rec__numeral_I2_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_num_int(numeral_numeral_int,Mb),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_num_int(numeral_numeral_int,Mb),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_num_int(numeral_numeral_int,Mb)),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_9615_upto__rec__numeral_I3_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_num_int(numeral_numeral_int,Nb)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_num_int(numeral_numeral_int,Nb)),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_num_int(numeral_numeral_int,Nb))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_9616_upto__rec__numeral_I4_J,axiom,
    ! [Mb: num,Nb: num] :
      upto(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb))),cons_int(aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb)),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Mb))),one_one_int),aa_int_int(uminus_uminus_int,aa_num_int(numeral_numeral_int,Nb)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_9617_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,J),upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K)) ) ) ) ).

% upto_split2
tff(fact_9618_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_9619_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or4662586982721622107an_int(I,J) = set_int2(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_9620_greaterThanAtMost__upto,axiom,
    ! [I: int,J: int] : set_or6656581121297822940st_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ).

% greaterThanAtMost_upto
tff(fact_9621_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)) ) ) ).

% upto_rec1
tff(fact_9622_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) ) ) ).

% upto.elims
tff(fact_9623_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J),cons_int(I,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),J)),nil_int) ).

% upto.simps
tff(fact_9624_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( upto(I,J) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_9625_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5832277885323065728an_int(I,J) = set_int2(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_9626_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( upto(I,K) = append_int(upto(I,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int),K))) ) ) ) ).

% upto_split3
tff(fact_9627_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list_int] :
      ( ( upto(Xb,Xaa) = Y )
     => ( accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xb,Xaa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Xb),Xaa),cons_int(Xb,upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb),one_one_int),Xaa)),nil_int) )
           => ~ accp_P1096762738010456898nt_int(upto_rel,product_Pair_int_int(Xb,Xaa)) ) ) ) ).

% upto.pelims
tff(fact_9628_list__encode_Opelims,axiom,
    ! [Xb: list_nat,Y: nat] :
      ( ( nat_list_encode(Xb) = Y )
     => ( aa_list_nat_o(accp_list_nat(nat_list_encode_rel),Xb)
       => ( ( ( Xb = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),nil_nat) ) )
         => ~ ! [X3: nat,Xs2: list_nat] :
                ( ( Xb = cons_nat(X3,Xs2) )
               => ( ( Y = aa_nat_nat(suc,aa_Pro4243173770044397973at_nat(nat_prod_encode,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X3),nat_list_encode(Xs2)))) )
                 => ~ aa_list_nat_o(accp_list_nat(nat_list_encode_rel),cons_nat(X3,Xs2)) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_9629_Rats__abs__iff,axiom,
    ! [Xb: real] :
      ( member_real(aa_real_real(abs_abs_real,Xb),field_5140801741446780682s_real)
    <=> member_real(Xb,field_5140801741446780682s_real) ) ).

% Rats_abs_iff
tff(fact_9630_Rats__no__top__le,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),X3) ) ).

% Rats_no_top_le
tff(fact_9631_Rats__no__bot__less,axiom,
    ! [Xb: real] :
    ? [X3: real] :
      ( member_real(X3,field_5140801741446780682s_real)
      & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Xb) ) ).

% Rats_no_bot_less
tff(fact_9632_Rats__dense__in__real,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [X3: real] :
          ( member_real(X3,field_5140801741446780682s_real)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),X3)
          & aa_real_o(aa_real_fun_real_o(ord_less_real,X3),Y) ) ) ).

% Rats_dense_in_real
tff(fact_9633_Rats__abs__nat__div__natE,axiom,
    ! [Xb: real] :
      ( member_real(Xb,field_5140801741446780682s_real)
     => ~ ! [M3: nat,N2: nat] :
            ( ( N2 != zero_zero_nat )
           => ( ( aa_real_real(abs_abs_real,Xb) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,M3)),aa_nat_real(semiri5074537144036343181t_real,N2)) )
             => ~ aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,M3),N2) ) ) ) ).

% Rats_abs_nat_div_natE
tff(fact_9634_upt__rec__numeral,axiom,
    ! [Mb: num,Nb: num] :
      upt(aa_num_nat(numeral_numeral_nat,Mb),aa_num_nat(numeral_numeral_nat,Nb)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb)),cons_nat(aa_num_nat(numeral_numeral_nat,Mb),upt(aa_nat_nat(suc,aa_num_nat(numeral_numeral_nat,Mb)),aa_num_nat(numeral_numeral_nat,Nb))),nil_nat) ).

% upt_rec_numeral
tff(fact_9635_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( hd_nat(upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_9636_drop__upt,axiom,
    ! [Mb: nat,I: nat,J: nat] : drop_nat(Mb,upt(I,J)) = upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb),J) ).

% drop_upt
tff(fact_9637_take__upt,axiom,
    ! [I: nat,Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)),Nb)
     => ( take_nat(Mb,upt(I,Nb)) = upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),Mb)) ) ) ).

% take_upt
tff(fact_9638_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I)
     => ( upt(I,J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_9639_length__upt,axiom,
    ! [I: nat,J: nat] : size_size_list_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I) ).

% length_upt
tff(fact_9640_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_9641_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K)),J)
     => ( aa_nat_nat(nth_nat(upt(I,J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),K) ) ) ).

% nth_upt
tff(fact_9642_sum__list__upt,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb)
     => ( groups4561878855575611511st_nat(upt(Mb,Nb)) = aa_set_nat_nat(groups3542108847815614940at_nat(aTP_Lamm_bb),aa_nat_set_nat(set_or4665077453230672383an_nat(Mb),Nb)) ) ) ).

% sum_list_upt
tff(fact_9643_atLeastAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(Nb),Mb) = set_nat2(upt(Nb,aa_nat_nat(suc,Mb))) ).

% atLeastAtMost_upt
tff(fact_9644_upt__conv__Cons__Cons,axiom,
    ! [Mb: nat,Nb: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(Mb,cons_nat(Nb,Ns)) = upt(Mb,Q2) )
    <=> ( cons_nat(Nb,Ns) = upt(aa_nat_nat(suc,Mb),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_9645_greaterThanAtMost__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),aa_nat_nat(suc,Mb))) ).

% greaterThanAtMost_upt
tff(fact_9646_greaterThanLessThan__upt,axiom,
    ! [Nb: nat,Mb: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(Nb),Mb) = set_nat2(upt(aa_nat_nat(suc,Nb),Mb)) ).

% greaterThanLessThan_upt
tff(fact_9647_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_9648_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( upt(I,J) = cons_nat(I,upt(aa_nat_nat(suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_9649_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(upt(I,J),upt(J,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_9650_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list_nat] :
      ( ( upt(I,J) = cons_nat(Xb,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
        & ( I = Xb )
        & ( upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I),one_one_nat),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_9651_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J),cons_nat(I,upt(aa_nat_nat(suc,I),J)),nil_nat) ).

% upt_rec
tff(fact_9652_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( upt(I,aa_nat_nat(suc,J)) = append_nat(upt(I,J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_9653_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J),append_nat(upt(I,J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_9654_mono__Suc,axiom,
    order_mono_nat_nat(suc) ).

% mono_Suc
tff(fact_9655_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_9656_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_hx,K)) ) ).

% mono_ge2_power_minus_self
tff(fact_9657_map__Suc__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(suc,upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb)) ).

% map_Suc_upt
tff(fact_9658_sorted__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),upt(Mb,Nb)) ).

% sorted_upt
tff(fact_9659_sorted__wrt__upt,axiom,
    ! [Mb: nat,Nb: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),upt(Mb,Nb)) ).

% sorted_wrt_upt
tff(fact_9660_map__add__upt,axiom,
    ! [Nb: nat,Mb: nat] : map_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hy,Nb),upt(zero_zero_nat,Mb)) = upt(Nb,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% map_add_upt
tff(fact_9661_map__decr__upt,axiom,
    ! [Mb: nat,Nb: nat] : map_nat_nat(aTP_Lamm_hz,upt(aa_nat_nat(suc,Mb),aa_nat_nat(suc,Nb))) = upt(Mb,Nb) ).

% map_decr_upt
tff(fact_9662_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),aa_nat_nat(nth_nat(Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_9663_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_9664_Rats__eq__int__div__int,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ia) ).

% Rats_eq_int_div_int
tff(fact_9665_Rats__eq__int__div__nat,axiom,
    field_5140801741446780682s_real = collect_real(aTP_Lamm_ib) ).

% Rats_eq_int_div_nat
tff(fact_9666_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_ic,Nb),top_top_set_nat) = aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),Nb) ) ) ).

% range_mod
tff(fact_9667_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_9668_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_9669_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_9670_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_9671_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa_real_real(aa_nat_fun_real_real(root,Nb),Xb) = $ite(Nb = zero_zero_nat,zero_zero_real,aa_real_real(the_in5290026491893676941l_real(top_top_set_real,aa_nat_fun_real_real(aTP_Lamm_id,Nb)),Xb)) ).

% root_def
tff(fact_9672_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_9673_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_9674_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_9675_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_9676_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_9677_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_9678_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(uminus_uminus_real,aa_nat_real(semiri5074537144036343181t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_even_real_root
tff(fact_9679_DERIV__real__root__generic,axiom,
    ! [Nb: nat,Xb: real,D4: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( Xb != zero_zero_real )
       => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
             => ( 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) ) )
         => ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
               => ( 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))))) ) ) )
           => ( ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
               => ( 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),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))) ) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_9680_DERIV__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_9681_DERIV__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_9682_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_9683_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2)),aa_real_real(F,A)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_9684_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2)),aa_real_real(F,A)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_9685_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_9686_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B2: real,G: fun_real_real,G2: fun_real_real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1222579329274155063t_real(A,B2))
         => has_fi5821293074295781190e_real(G,aa_real_real(G2,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ( ! [X3: real] :
            ( member_real(X3,set_or1222579329274155063t_real(A,B2))
           => 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),B2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(G,A)),aa_real_real(G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_9687_DERIV__isconst__all,axiom,
    ! [F: fun_real_real,Xb: real,Y: real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real))
     => ( aa_real_real(F,Xb) = aa_real_real(F,Y) ) ) ).

% DERIV_isconst_all
tff(fact_9688_DERIV__mirror,axiom,
    ! [F: fun_real_real,Y: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,Y,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,Xb),top_top_set_real))
    <=> has_fi5821293074295781190e_real(aTP_Lamm_ie(F),aa_real_real(uminus_uminus_real,Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_mirror
tff(fact_9689_DERIV__const__ratio__const2,axiom,
    ! [A: real,B2: real,F: fun_real_real,K: real] :
      ( ( A != B2 )
     => ( ! [X3: real] : has_fi5821293074295781190e_real(F,K,topolo2177554685111907308n_real(X3,top_top_set_real))
       => ( aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A))),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_9690_DERIV__const__ratio__const,axiom,
    ! [A: real,B2: real,F: fun_real_real,K: real] :
      ( ( A != B2 )
     => ( ! [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,B2)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_9691_DERIV__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_9692_DERIV__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                 => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_9693_DERIV__isconst3,axiom,
    ! [A: real,B2: real,Xb: real,Y: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( member_real(Xb,set_or1633881224788618240n_real(A,B2))
       => ( member_real(Y,set_or1633881224788618240n_real(A,B2))
         => ( ! [X3: real] :
                ( member_real(X3,set_or1633881224788618240n_real(A,B2))
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) )
           => ( aa_real_real(F,Xb) = aa_real_real(F,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_9694_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_9695_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_9696_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,Xb)),aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_9697_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun_real_real,L: real,Xb: real,S2: set_real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,S2))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [D2: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D2)
            & ! [H3: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),H3)
               => ( member_real(aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3),S2)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,H3),D2)
                   => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),H3))),aa_real_real(F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_9698_MVT2,axiom,
    ! [A: real,B2: real,F: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2)
             => has_fi5821293074295781190e_real(F,aa_real_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
       => ? [Z: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
            & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
            & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),aa_real_real(F3,Z)) ) ) ) ) ).

% MVT2
tff(fact_9699_DERIV__local__const,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => ( aa_real_real(F,Xb) = aa_real_real(F,Y3) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
tff(fact_9700_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(inverse_inverse_real,Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln
tff(fact_9701_DERIV__const__average,axiom,
    ! [A: real,B2: real,V: fun_real_real,K: real] :
      ( ( A != B2 )
     => ( ! [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),B2)),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,B2))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) ) ) ).

% DERIV_const_average
tff(fact_9702_DERIV__local__max,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Y3)),aa_real_real(F,Xb)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
tff(fact_9703_DERIV__local__min,axiom,
    ! [F: fun_real_real,L: real,Xb: real,D: real] :
      ( has_fi5821293074295781190e_real(F,L,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
       => ( ! [Y3: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y3))),D)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,Xb)),aa_real_real(F,Y3)) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
tff(fact_9704_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(ln_ln_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Xb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_ln_divide
tff(fact_9705_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S: set_real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_if,Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat)))),topolo2177554685111907308n_real(Xb,S)) ).

% DERIV_pow
tff(fact_9706_DERIV__fun__pow,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,Nb: nat] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ig(G),Nb),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(G,Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),one_one_nat)))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_fun_pow
tff(fact_9707_has__real__derivative__powr,axiom,
    ! [Zb: real,R2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Zb)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ih,R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,Zb),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),one_one_real))),topolo2177554685111907308n_real(Zb,top_top_set_real)) ) ).

% has_real_derivative_powr
tff(fact_9708_DERIV__log,axiom,
    ! [Xb: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(log,B2),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,B2)),Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_log
tff(fact_9709_DERIV__fun__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => has_fi5821293074295781190e_real(aa_rea7340605660300375871l_real(aTP_Lamm_ii(G),R2),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,R2),aa_nat_real(semiri5074537144036343181t_real,one_one_nat))))),Mb),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_fun_powr
tff(fact_9710_DERIV__powr,axiom,
    ! [G: fun_real_real,Mb: real,Xb: real,F: fun_real_real,R2: real] :
      ( has_fi5821293074295781190e_real(G,Mb,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(G,Xb))
       => ( has_fi5821293074295781190e_real(F,R2,topolo2177554685111907308n_real(Xb,top_top_set_real))
         => has_fi5821293074295781190e_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(G),F),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(G,Xb)),aa_real_real(F,Xb))),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(ln_ln_real,aa_real_real(G,Xb)))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Mb),aa_real_real(F,Xb))),aa_real_real(G,Xb)))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_powr
tff(fact_9711_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
     => has_fi5821293074295781190e_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_real_sqrt
tff(fact_9712_DERIV__series_H,axiom,
    ! [F: fun_re4238438383274954274t_real,F3: fun_re4238438383274954274t_real,X0: real,A: real,B2: real,L5: fun_nat_real] :
      ( ! [N2: nat] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(aTP_Lamm_ik(F),N2),aa_nat_real(aa_real_fun_nat_real(F3,X0),N2),topolo2177554685111907308n_real(X0,top_top_set_real))
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(A,B2))
           => summable_real(aa_real_fun_nat_real(F,X3)) )
       => ( member_real(X0,set_or1633881224788618240n_real(A,B2))
         => ( summable_real(aa_real_fun_nat_real(F3,X0))
           => ( summable_real(L5)
             => ( ! [N2: nat,X3: real,Y3: real] :
                    ( member_real(X3,set_or1633881224788618240n_real(A,B2))
                   => ( member_real(Y3,set_or1633881224788618240n_real(A,B2))
                     => 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),N2)),aa_nat_real(aa_real_fun_nat_real(F,Y3),N2)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(L5,N2)),aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,X3),Y3)))) ) )
               => has_fi5821293074295781190e_real(aTP_Lamm_il(F),suminf_real(aa_real_fun_nat_real(F3,X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_9713_DERIV__arctan,axiom,
    ! [Xb: real] : has_fi5821293074295781190e_real(arctan,aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ).

% DERIV_arctan
tff(fact_9714_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] : has_fi5821293074295781190e_real(arsinh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_9715_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D4: real] :
      ( ( Xb != zero_zero_real )
     => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
         => ( D4 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,Xb))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
       => ( ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),zero_zero_real)
           => ( 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,Xb)))),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))) ) )
         => has_fi5821293074295781190e_real(sqrt,D4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_9716_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => has_fi5821293074295781190e_real(arcosh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_real))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_9717_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A3: set_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(artanh_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),topolo2177554685111907308n_real(Xb,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_9718_DERIV__power__series_H,axiom,
    ! [R: real,F: fun_nat_real,X0: real] :
      ( ! [X3: real] :
          ( member_real(X3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
         => summable_real(aa_real_fun_nat_real(aTP_Lamm_im(F),X3)) )
     => ( member_real(X0,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,R),R))
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R)
         => has_fi5821293074295781190e_real(aTP_Lamm_io(F),suminf_real(aa_real_fun_nat_real(aTP_Lamm_im(F),X0)),topolo2177554685111907308n_real(X0,top_top_set_real)) ) ) ) ).

% DERIV_power_series'
tff(fact_9719_DERIV__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),Xb)
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_real_root
tff(fact_9720_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arccos,aa_real_real(inverse_inverse_real,aa_real_real(uminus_uminus_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arccos
tff(fact_9721_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => has_fi5821293074295781190e_real(arcsin,aa_real_real(inverse_inverse_real,aa_real_real(sqrt,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_arcsin
tff(fact_9722_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
        & ! [M3: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X3),topolo2177554685111907308n_real(X3,top_top_set_real)) )
     => ? [T: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
          & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ip(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_9723_Maclaurin__all__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Xb: real,Nb: nat] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ip(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_9724_DERIV__odd__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
     => ( ( Xb != zero_zero_real )
       => has_fi5821293074295781190e_real(aa_nat_fun_real_real(root,Nb),aa_real_real(inverse_inverse_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_nat_fun_real_real(root,Nb),Xb)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,zero_zero_nat))))),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ).

% DERIV_odd_real_root
tff(fact_9725_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 )
         => ( ! [M3: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),H)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iq(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),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_9726_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 )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
         => ? [T: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),T)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H)
              & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iq(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),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_9727_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 )
         => ( ! [M3: nat,T: real] :
                ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,H),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),zero_zero_real) )
               => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,H),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,T),zero_zero_real)
                & ( aa_real_real(F,H) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_fun1113820866546176217t_real(aTP_Lamm_iq(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),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,H),Nb))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_9728_Maclaurin__all__lt,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
       => ( ( Xb != zero_zero_real )
         => ( ! [M3: nat,X3: real] : has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),X3),topolo2177554685111907308n_real(X3,top_top_set_real))
           => ? [T: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(abs_abs_real,T))
                & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
                & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ip(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_9729_Maclaurin__bi__le,axiom,
    ! [Diff: fun_na6724076748692969250l_real,F: fun_real_real,Nb: nat,Xb: real] :
      ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
     => ( ! [M3: nat,T: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb)) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
       => ? [T: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,T)),aa_real_real(abs_abs_real,Xb))
            & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aTP_Lamm_ip(Diff),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Xb),Nb))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_9730_Taylor,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: real,C: real,Xb: real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( ( aa_nat_fun_real_real(Diff,zero_zero_nat) = F )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,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),B2)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Xb)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B2)
                 => ( ( Xb != C )
                   => ? [T: real] :
                        ( $ite(
                            aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),C),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),C) ),
                            ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                            & aa_real_o(aa_real_fun_real_o(ord_less_real,T),Xb) ) )
                        & ( aa_real_real(F,Xb) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ir(Diff),C),Xb),aa_nat_set_nat(set_ord_lessThan_nat,Nb))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Diff,Nb),T)),aa_nat_real(semiri2265585572941072030t_real,Nb))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),C)),Nb))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_9731_Taylor__up,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: 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 )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,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),B2)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,C),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),B2)
                  & ( aa_real_real(F,B2) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_is(Diff),B2),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),T)),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,B2),C)),Nb))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_9732_Taylor__down,axiom,
    ! [Nb: nat,Diff: fun_na6724076748692969250l_real,F: fun_real_real,A: real,B2: 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 )
       => ( ! [M3: nat,T: real] :
              ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),T)
                & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),B2) )
             => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,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),B2)
             => ? [T: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),T)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,T),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_is(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),T)),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_9733_Maclaurin__lemma2,axiom,
    ! [Nb: nat,H: real,Diff: fun_na6724076748692969250l_real,K: nat,B3: real] :
      ( ! [M3: nat,T: real] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M3),Nb)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T)
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T),H) )
         => has_fi5821293074295781190e_real(aa_nat_fun_real_real(Diff,M3),aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M3)),T),topolo2177554685111907308n_real(T,top_top_set_real)) )
     => ( ( Nb = aa_nat_nat(suc,K) )
       => ! [M4: nat,T2: real] :
            ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),Nb)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),T2)
              & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,T2),H) )
           => has_fi5821293074295781190e_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_iu(Nb),Diff),B3),M4),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Diff,aa_nat_nat(suc,M4)),T2)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_iv(Diff),M4),T2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M4))))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,B3),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,T2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M4)))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Nb),aa_nat_nat(suc,M4))))))),topolo2177554685111907308n_real(T2,top_top_set_real)) ) ) ) ).

% Maclaurin_lemma2
tff(fact_9734_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => has_fi5821293074295781190e_real(aTP_Lamm_iw,suminf_real(aa_real_fun_nat_real(aTP_Lamm_ix,Xb)),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ).

% DERIV_arctan_series
tff(fact_9735_isCont__Lb__Ub,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( ! [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),B2) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(X3,top_top_set_real),F) )
       => ? [L6: real,M8: real] :
            ( ! [X4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),X4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X4),B2) )
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_real_real(F,X4))
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(F,X4)),M8) ) )
            & ! [Y4: real] :
                ( ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),Y4)
                  & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Y4),M8) )
               => ? [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),B2)
                    & ( aa_real_real(F,X3) = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_9736_LIM__fun__less__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,L),zero_zero_real)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,X4)),zero_zero_real) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_9737_LIM__fun__not__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( ( L != zero_zero_real )
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3) )
               => ( aa_real_real(F,X4) != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_9738_LIM__fun__gt__zero,axiom,
    ! [F: fun_real_real,L: real,C: real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(L),topolo2177554685111907308n_real(C,top_top_set_real))
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),L)
       => ? [R3: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),R3)
            & ! [X4: real] :
                ( ( ( X4 != 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),X4))),R3) )
               => aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),aa_real_real(F,X4)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_9739_isCont__real__sqrt,axiom,
    ! [Xb: real] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),sqrt) ).

% isCont_real_sqrt
tff(fact_9740_isCont__real__root,axiom,
    ! [Xb: real,Nb: nat] : topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),aa_nat_fun_real_real(root,Nb)) ).

% isCont_real_root
tff(fact_9741_continuous__frac,axiom,
    ! [Xb: real] :
      ( ~ member_real(Xb,ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),archim2898591450579166408c_real) ) ).

% continuous_frac
tff(fact_9742_isCont__inverse__function2,axiom,
    ! [A: real,Xb: real,B2: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
       => ( ! [Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B2)
               => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) ) )
         => ( ! [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B2)
                 => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
           => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_9743_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcosh_real) ) ).

% isCont_arcosh
tff(fact_9744_LIM__cos__div__sin,axiom,
    filterlim_real_real(aTP_Lamm_iy,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_9745_continuous__floor,axiom,
    ! [Xb: real] :
      ( ~ member_real(Xb,ring_1_Ints_real)
     => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),comp_int_real_real(ring_1_of_int_real,archim6058952711729229775r_real)) ) ).

% continuous_floor
tff(fact_9746_DERIV__inverse__function,axiom,
    ! [F: fun_real_real,D4: real,G: fun_real_real,Xb: real,A: real,B2: real] :
      ( has_fi5821293074295781190e_real(F,D4,topolo2177554685111907308n_real(aa_real_real(G,Xb),top_top_set_real))
     => ( ( D4 != zero_zero_real )
       => ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xb)
         => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),B2)
           => ( ! [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),B2)
                   => ( aa_real_real(F,aa_real_real(G,Y3)) = Y3 ) ) )
             => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),G)
               => has_fi5821293074295781190e_real(G,aa_real_real(inverse_inverse_real,D4),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_9747_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arccos) ) ) ).

% isCont_arccos
tff(fact_9748_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),arcsin) ) ) ).

% isCont_arcsin
tff(fact_9749_LIM__less__bound,axiom,
    ! [B2: real,Xb: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),Xb)
     => ( ! [X3: real] :
            ( member_real(X3,set_or1633881224788618240n_real(B2,Xb))
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,X3)) )
       => ( topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_real_real(F,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_9750_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,one_one_real)),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Xb,top_top_set_real),artanh_real) ) ) ).

% isCont_artanh
tff(fact_9751_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun_real_real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),D)
     => ( ! [Z: 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,Z),Xb))),D)
           => ( aa_real_real(G,aa_real_real(F,Z)) = Z ) )
       => ( ! [Z: 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,Z),Xb))),D)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) )
         => topolo4422821103128117721l_real(topolo2177554685111907308n_real(aa_real_real(F,Xb),top_top_set_real),G) ) ) ) ).

% isCont_inverse_function
tff(fact_9752_GMVT_H,axiom,
    ! [A: real,B2: real,F: fun_real_real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [Z: real] :
            ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B2)
             => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),F) ) )
       => ( ! [Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),Z)
             => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Z),B2)
               => topolo4422821103128117721l_real(topolo2177554685111907308n_real(Z,top_top_set_real),G) ) )
         => ( ! [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
                 => has_fi5821293074295781190e_real(G,aa_real_real(G2,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
           => ( ! [Z: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
                 => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
                   => has_fi5821293074295781190e_real(F,aa_real_real(F3,Z),topolo2177554685111907308n_real(Z,top_top_set_real)) ) )
             => ? [C3: real] :
                  ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A))),aa_real_real(G2,C3)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(G,B2)),aa_real_real(G,A))),aa_real_real(F3,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_9753_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] : member_real(suminf_real(aTP_Lamm_iz(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_iz(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_iz(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_9754_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] : member_real(suminf_real(aTP_Lamm_iz(A)),set_or1222579329274155063t_real(groups6591440286371151544t_real(aTP_Lamm_iz(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_iz(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_9755_incseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I3)),B3)
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I4)),L6) ) ) ) ).

% incseq_convergent
tff(fact_9756_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_9757_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_aw,C),at_top_nat,at_top_nat) ) ).

% mult_nat_right_at_top
tff(fact_9758_monoseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( topolo6980174941875973593q_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,aa_nat_real(X5,I3))),B3)
       => ~ ! [L6: real] : ~ filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat) ) ) ).

% monoseq_convergent
tff(fact_9759_LIMSEQ__root,axiom,
    filterlim_nat_real(aTP_Lamm_ja,topolo2815343760600316023s_real(one_one_real),at_top_nat) ).

% LIMSEQ_root
tff(fact_9760_nested__sequence__unique,axiom,
    ! [F: fun_nat_real,G: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(G,aa_nat_nat(suc,N2))),aa_nat_real(G,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(G,N2))
         => ( filterlim_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jb(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_9761_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun_nat_real] :
      ( ! [R3: real] :
        ? [N6: nat] :
        ! [N2: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N6),N2)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,R3),aa_nat_real(X5,N2)) )
     => filterlim_nat_real(aTP_Lamm_jc(X5),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_zero
tff(fact_9762_lim__inverse__n_H,axiom,
    filterlim_nat_real(aTP_Lamm_jd,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% lim_inverse_n'
tff(fact_9763_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_je,C),topolo2815343760600316023s_real(one_one_real),at_top_nat) ) ).

% LIMSEQ_root_const
tff(fact_9764_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim_nat_real(aTP_Lamm_jf,topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_9765_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jg,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_9766_increasing__LIMSEQ,axiom,
    ! [F: fun_nat_real,L: real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,aa_nat_nat(suc,N2)))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),L)
       => ( ! [E2: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,zero_zero_real),E2)
             => ? [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)),E2)) )
         => filterlim_nat_real(F,topolo2815343760600316023s_real(L),at_top_nat) ) ) ) ).

% increasing_LIMSEQ
tff(fact_9767_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),one_one_real)
       => filterlim_nat_real(aa_real_fun_nat_real(power_power_real,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_9768_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jh(Xb),A),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_9769_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_9770_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_9771_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),Xb)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_ji,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_9772_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jj,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_9773_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jk,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_nat) ).

% tendsto_exp_limit_sequentially
tff(fact_9774_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jl,R2),topolo2815343760600316023s_real(R2),at_top_nat) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_9775_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_iz(A)) ) ) ).

% summable_Leibniz(1)
tff(fact_9776_summable,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => summable_real(aTP_Lamm_iz(A)) ) ) ) ).

% summable
tff(fact_9777_cos__diff__limit__1,axiom,
    ! [Theta: fun_nat_real,Theta2: real] :
      ( filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_jm(Theta),Theta2),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ~ ! [K2: fun_nat_int] : ~ filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_jn(Theta),K2),topolo2815343760600316023s_real(Theta2),at_top_nat) ) ).

% cos_diff_limit_1
tff(fact_9778_cos__limit__1,axiom,
    ! [Theta: fun_nat_real] :
      ( filterlim_nat_real(aTP_Lamm_jo(Theta),topolo2815343760600316023s_real(one_one_real),at_top_nat)
     => ? [K2: fun_nat_int] : filterlim_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_jn(Theta),K2),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% cos_limit_1
tff(fact_9779_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_jp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_iz(A))),at_top_nat) ) ) ).

% summable_Leibniz(4)
tff(fact_9780_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(abs_abs_real,Xb)),one_one_real)
     => filterlim_nat_real(aa_real_fun_nat_real(aTP_Lamm_bt,Xb),topolo2815343760600316023s_real(zero_zero_real),at_top_nat) ) ).

% zeroseq_arctan_series
tff(fact_9781_summable__Leibniz_H_I2_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,groups6591440286371151544t_real(aTP_Lamm_iz(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_iz(A))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_9782_summable__Leibniz_H_I3_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_jp(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_iz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_9783_sums__alternating__upper__lower,axiom,
    ! [A: fun_nat_real] :
      ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( 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_iz(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_jp(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_iz(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_jq(A),topolo2815343760600316023s_real(L2),at_top_nat) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_9784_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_jq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_iz(A))),at_top_nat) ) ) ).

% summable_Leibniz(5)
tff(fact_9785_summable__Leibniz_H_I5_J,axiom,
    ! [A: fun_nat_real] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => filterlim_nat_real(aTP_Lamm_jq(A),topolo2815343760600316023s_real(suminf_real(aTP_Lamm_iz(A))),at_top_nat) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_9786_summable__Leibniz_H_I4_J,axiom,
    ! [A: fun_nat_real,Nb: nat] :
      ( filterlim_nat_real(A,topolo2815343760600316023s_real(zero_zero_real),at_top_nat)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(A,N2))
       => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(A,aa_nat_nat(suc,N2))),aa_nat_real(A,N2))
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,suminf_real(aTP_Lamm_iz(A))),groups6591440286371151544t_real(aTP_Lamm_iz(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_9787_eventually__sequentially__Suc,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(aTP_Lamm_jr(P),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_Suc
tff(fact_9788_eventually__sequentially__seg,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_js(P),K),at_top_nat)
    <=> eventually_nat(P,at_top_nat) ) ).

% eventually_sequentially_seg
tff(fact_9789_sequentially__offset,axiom,
    ! [P: fun_nat_o,K: nat] :
      ( eventually_nat(P,at_top_nat)
     => eventually_nat(aa_nat_fun_nat_o(aTP_Lamm_js(P),K),at_top_nat) ) ).

% sequentially_offset
tff(fact_9790_eventually__sequentially,axiom,
    ! [P: fun_nat_o] :
      ( eventually_nat(P,at_top_nat)
    <=> ? [N7: nat] :
        ! [N4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N4)
         => aa_nat_o(P,N4) ) ) ).

% eventually_sequentially
tff(fact_9791_eventually__sequentiallyI,axiom,
    ! [C: nat,P: fun_nat_o] :
      ( ! [X3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),X3)
         => aa_nat_o(P,X3) )
     => eventually_nat(P,at_top_nat) ) ).

% eventually_sequentiallyI
tff(fact_9792_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ord_le2510731241096832064er_nat(F4,at_top_nat)
    <=> ! [N7: nat] : eventually_nat(aa_nat_fun_nat_o(ord_less_eq_nat,N7),F4) ) ).

% le_sequentially
tff(fact_9793_filterlim__Suc,axiom,
    filterlim_nat_nat(suc,at_top_nat,at_top_nat) ).

% filterlim_Suc
tff(fact_9794_real__bounded__linear,axiom,
    ! [F: fun_real_real] :
      ( real_V5970128139526366754l_real(F)
    <=> ? [C2: real] :
        ! [X: real] : aa_real_real(F,X) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,X),C2) ) ).

% real_bounded_linear
tff(fact_9795_sqrt__at__top,axiom,
    filterlim_real_real(sqrt,at_top_real,at_top_real) ).

% sqrt_at_top
tff(fact_9796_lhopital__left__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_9797_lhopital__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_top_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_9798_eventually__at__left__real,axiom,
    ! [B2: real,A: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,B2),A)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_jv(B2),A),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ).

% eventually_at_left_real
tff(fact_9799_lhopital__left__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_9800_lhospital__at__top__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,at_top_real)
     => ( eventually_real(aTP_Lamm_jw(G2),at_top_real)
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),at_top_real)
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),at_top_real)
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),topolo2815343760600316023s_real(Xb),at_top_real)
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G),F),topolo2815343760600316023s_real(Xb),at_top_real) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_9801_lhopital__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_9802_tanh__real__at__top,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(one_one_real),at_top_real) ).

% tanh_real_at_top
tff(fact_9803_artanh__real__at__left__1,axiom,
    filterlim_real_real(artanh_real,at_top_real,topolo2177554685111907308n_real(one_one_real,set_or5984915006950818249n_real(one_one_real))) ).

% artanh_real_at_left_1
tff(fact_9804_ln__x__over__x__tendsto__0,axiom,
    filterlim_real_real(aTP_Lamm_jy,topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% ln_x_over_x_tendsto_0
tff(fact_9805_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim_real_real(aa_nat_fun_real_real(aTP_Lamm_jz,K),topolo2815343760600316023s_real(zero_zero_real),at_top_real) ).

% tendsto_power_div_exp_0
tff(fact_9806_lhopital__left,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
       => ( eventually_real(aTP_Lamm_jw(G),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
         => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),F4,topolo2177554685111907308n_real(Xb,set_or5984915006950818249n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_9807_lhopital,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,top_top_set_real))
       => ( eventually_real(aTP_Lamm_jw(G),topolo2177554685111907308n_real(Xb,top_top_set_real))
         => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,top_top_set_real))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,top_top_set_real))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),F4,topolo2177554685111907308n_real(Xb,top_top_set_real))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),F4,topolo2177554685111907308n_real(Xb,top_top_set_real)) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_9808_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ka,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),at_top_real) ).

% tendsto_exp_limit_at_top
tff(fact_9809_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_9810_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),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,B2)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_9811_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_9812_filterlim__pow__at__bot__even,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kb(Nb),F),at_top_real,F4) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_9813_dist__real__def,axiom,
    ! [Xb: real,Y: real] : aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Xb),Y) = aa_real_real(abs_abs_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Xb),Y)) ).

% dist_real_def
tff(fact_9814_dist__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa_complex_real(real_V3694042436643373181omplex(Xb),Y) = aa_complex_real(real_V1022390504157884413omplex,aa_complex_complex(minus_minus_complex(Xb),Y)) ).

% dist_complex_def
tff(fact_9815_tanh__real__at__bot,axiom,
    filterlim_real_real(tanh_real,topolo2815343760600316023s_real(aa_real_real(uminus_uminus_real,one_one_real)),at_bot_real) ).

% tanh_real_at_bot
tff(fact_9816_lhopital__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,top_top_set_real))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,top_top_set_real))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,top_top_set_real))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_bot_real,topolo2177554685111907308n_real(A,top_top_set_real)) ) ) ) ) ) ).

% lhopital_at_top_at_bot
tff(fact_9817_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F: fun_real_real,Flim: real] :
      ( ! [X3: real] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X3),B2)
         => ? [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,B2)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_9818_filterlim__pow__at__bot__odd,axiom,
    ! [Nb: nat,F: fun_real_real,F4: filter_real] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => ( filterlim_real_real(F,at_bot_real,F4)
       => ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb)
         => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kb(Nb),F),at_bot_real,F4) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_9819_lhopital__left__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5984915006950818249n_real(A))) ) ) ) ) ) ).

% lhopital_left_at_top_at_bot
tff(fact_9820_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_9821_incseq__bounded,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_mono_nat_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,I3)),B3)
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% incseq_bounded
tff(fact_9822_Bseq__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => bfun_nat_real(aa_real_fun_nat_real(power_power_real,Xb),at_top_nat) ) ) ).

% Bseq_realpow
tff(fact_9823_tendsto__exp__limit__at__right,axiom,
    ! [Xb: real] : filterlim_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_kc,Xb),topolo2815343760600316023s_real(aa_real_real(exp_real,Xb)),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% tendsto_exp_limit_at_right
tff(fact_9824_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_9825_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_kd(P),A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ).

% eventually_at_right_to_0
tff(fact_9826_eventually__at__right__real,axiom,
    ! [A: real,B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => eventually_real(aa_real_fun_real_o(aTP_Lamm_jv(A),B2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ).

% eventually_at_right_real
tff(fact_9827_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_9828_artanh__real__at__right__1,axiom,
    filterlim_real_real(artanh_real,at_bot_real,topolo2177554685111907308n_real(aa_real_real(uminus_uminus_real,one_one_real),set_or5849166863359141190n_real(aa_real_real(uminus_uminus_real,one_one_real)))) ).

% artanh_real_at_right_1
tff(fact_9829_lhopital__right__at__top__at__top,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_top
tff(fact_9830_lhopital__right__0,axiom,
    ! [F0: fun_real_real,G0: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: 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_jw(G0),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F0),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G0),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F0),G0),F4,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ) ) ).

% lhopital_right_0
tff(fact_9831_lhopital__right,axiom,
    ! [F: fun_real_real,Xb: real,G: fun_real_real,G2: fun_real_real,F3: fun_real_real,F4: filter_real] :
      ( filterlim_real_real(F,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( filterlim_real_real(G,topolo2815343760600316023s_real(zero_zero_real),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aTP_Lamm_jw(G),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
               => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),F4,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
                 => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),F4,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ) ) ).

% lhopital_right
tff(fact_9832_lhopital__right__at__top__at__bot,axiom,
    ! [F: fun_real_real,A: real,G: fun_real_real,F3: fun_real_real,G2: fun_real_real] :
      ( filterlim_real_real(F,at_top_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
     => ( filterlim_real_real(G,at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F3),G2),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(F),G),at_bot_real,topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) ) ) ) ) ) ).

% lhopital_right_at_top_at_bot
tff(fact_9833_lhopital__right__at__top,axiom,
    ! [G: fun_real_real,Xb: real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Y: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
     => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G),F),topolo2815343760600316023s_real(Y),topolo2177554685111907308n_real(Xb,set_or5849166863359141190n_real(Xb))) ) ) ) ) ) ).

% lhopital_right_at_top
tff(fact_9834_lhopital__right__0__at__top,axiom,
    ! [G: fun_real_real,G2: fun_real_real,F: fun_real_real,F3: fun_real_real,Xb: real] :
      ( filterlim_real_real(G,at_top_real,topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
     => ( eventually_real(aTP_Lamm_jw(G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
       => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(F),F3),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
         => ( eventually_real(aa_fun2895569901277098738real_o(aTP_Lamm_jt(G),G2),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
           => ( filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G2),F3),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real)))
             => filterlim_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(G),F),topolo2815343760600316023s_real(Xb),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ) ) ) ) ) ).

% lhopital_right_0_at_top
tff(fact_9835_atLeast__0,axiom,
    aa_nat_set_nat(set_ord_atLeast_nat,zero_zero_nat) = top_top_set_nat ).

% atLeast_0
tff(fact_9836_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_9837_decseq__bounded,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X5,I3))
       => bfun_nat_real(X5,at_top_nat) ) ) ).

% decseq_bounded
tff(fact_9838_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_9839_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_9840_decseq__convergent,axiom,
    ! [X5: fun_nat_real,B3: real] :
      ( order_9091379641038594480t_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B3),aa_nat_real(X5,I3))
       => ~ ! [L6: real] :
              ( filterlim_nat_real(X5,topolo2815343760600316023s_real(L6),at_top_nat)
             => ~ ! [I4: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,L6),aa_nat_real(X5,I4)) ) ) ) ).

% decseq_convergent
tff(fact_9841_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_9842_GMVT,axiom,
    ! [A: real,B2: real,F: fun_real_real,G: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2) )
           => 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),B2) )
             => 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),B2) )
               => 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),B2) )
                 => differ6690327859849518006l_real(G,topolo2177554685111907308n_real(X3,top_top_set_real)) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( has_fi5821293074295781190e_real(G,G_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & has_fi5821293074295781190e_real(F,F_c,topolo2177554685111907308n_real(C3,top_top_set_real))
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,A),C3)
                  & aa_real_o(aa_real_fun_real_o(ord_less_real,C3),B2)
                  & ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),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,B2)),aa_real_real(G,A))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_9843_real__differentiableE,axiom,
    ! [F: fun_real_real,Xb: real,S: set_real] :
      ( differ6690327859849518006l_real(F,topolo2177554685111907308n_real(Xb,S))
     => ~ ! [Df: real] : ~ has_fi5821293074295781190e_real(F,Df,topolo2177554685111907308n_real(Xb,S)) ) ).

% real_differentiableE
tff(fact_9844_real__differentiable__def,axiom,
    ! [F: fun_real_real,Xb: real,S: set_real] :
      ( differ6690327859849518006l_real(F,topolo2177554685111907308n_real(Xb,S))
    <=> ? [D6: real] : has_fi5821293074295781190e_real(F,D6,topolo2177554685111907308n_real(Xb,S)) ) ).

% real_differentiable_def
tff(fact_9845_MVT,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
               => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ? [L2: real,Z: real] :
              ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
              & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
              & has_fi5821293074295781190e_real(F,L2,topolo2177554685111907308n_real(Z,top_top_set_real))
              & ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)),L2) ) ) ) ) ) ).

% MVT
tff(fact_9846_suminf__eq__SUP__real,axiom,
    ! [X5: fun_nat_real] :
      ( summable_real(X5)
     => ( ! [I3: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),aa_nat_real(X5,I3))
       => ( suminf_real(X5) = comple1385675409528146559p_real(image_nat_real(aTP_Lamm_ke(X5),top_top_set_nat)) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_9847_continuous__on__arcosh,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,set_ord_atLeast_real(one_one_real))
     => topolo5044208981011980120l_real(A3,arcosh_real) ) ).

% continuous_on_arcosh
tff(fact_9848_continuous__on__arcosh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X3: real] :
            ( member_real(X3,A3)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,one_one_real),aa_real_real(F,X3)) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_kf(F)) ) ) ).

% continuous_on_arcosh'
tff(fact_9849_continuous__image__closed__interval,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),F)
       => ? [C3: real,D2: real] :
            ( ( image_real_real(F,set_or1222579329274155063t_real(A,B2)) = set_or1222579329274155063t_real(C3,D2) )
            & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,C3),D2) ) ) ) ).

% continuous_image_closed_interval
tff(fact_9850_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arccos) ).

% continuous_on_arccos'
tff(fact_9851_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real(set_or1222579329274155063t_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real),arcsin) ).

% continuous_on_arcsin'
tff(fact_9852_continuous__on__artanh,axiom,
    ! [A3: set_real] :
      ( ord_less_eq_set_real(A3,set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real))
     => topolo5044208981011980120l_real(A3,artanh_real) ) ).

% continuous_on_artanh
tff(fact_9853_continuous__on__artanh_H,axiom,
    ! [A3: set_real,F: fun_real_real] :
      ( topolo5044208981011980120l_real(A3,F)
     => ( ! [X3: real] :
            ( member_real(X3,A3)
           => member_real(aa_real_real(F,X3),set_or1633881224788618240n_real(aa_real_real(uminus_uminus_real,one_one_real),one_one_real)) )
       => topolo5044208981011980120l_real(A3,aTP_Lamm_kg(F)) ) ) ).

% continuous_on_artanh'
tff(fact_9854_Rolle__deriv,axiom,
    ! [A: real,B2: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
                 => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
                & ! [X4: real] : aa_real_real(aa_rea7340605660300375871l_real(F3,Z),X4) = zero_zero_real ) ) ) ) ) ).

% Rolle_deriv
tff(fact_9855_mvt,axiom,
    ! [A: real,B2: real,F: fun_real_real,F3: fun_re7681423839302404990l_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
               => has_de1759254742604945161l_real(F,aa_rea7340605660300375871l_real(F3,X3),topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ~ ! [Xi: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Xi)
               => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xi),B2)
                 => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(F,B2)),aa_real_real(F,A)) != aa_real_real(aa_rea7340605660300375871l_real(F3,Xi),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,B2),A)) ) ) ) ) ) ) ).

% mvt
tff(fact_9856_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,A)),aa_real_real(F,B2)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_9857_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ! [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),B2)
             => ? [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,B2),F)
         => aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(F,B2)),aa_real_real(F,A)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_9858_DERIV__isconst__end,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
               => has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
         => ( aa_real_real(F,B2) = aa_real_real(F,A) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_9859_DERIV__isconst2,axiom,
    ! [A: real,B2: real,F: fun_real_real,Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
               => 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),Xb)
           => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),B2)
             => ( aa_real_real(F,Xb) = aa_real_real(F,A) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_9860_Rolle,axiom,
    ! [A: real,B2: real,F: fun_real_real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),B2)
     => ( ( aa_real_real(F,A) = aa_real_real(F,B2) )
       => ( topolo5044208981011980120l_real(set_or1222579329274155063t_real(A,B2),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),B2)
                 => differ6690327859849518006l_real(F,topolo2177554685111907308n_real(X3,top_top_set_real)) ) )
           => ? [Z: real] :
                ( aa_real_o(aa_real_fun_real_o(ord_less_real,A),Z)
                & aa_real_o(aa_real_fun_real_o(ord_less_real,Z),B2)
                & has_fi5821293074295781190e_real(F,zero_zero_real,topolo2177554685111907308n_real(Z,top_top_set_real)) ) ) ) ) ) ).

% Rolle
tff(fact_9861_UN__lessThan__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_lessThan_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_lessThan_UNIV
tff(fact_9862_UN__atMost__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atMost_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atMost_UNIV
tff(fact_9863_UN__atLeast__UNIV,axiom,
    comple7399068483239264473et_nat(image_nat_set_nat(set_ord_atLeast_nat,top_top_set_nat)) = top_top_set_nat ).

% UN_atLeast_UNIV
tff(fact_9864_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_9865_Inf__real__def,axiom,
    ! [X5: set_real] : comple4887499456419720421f_real(X5) = aa_real_real(uminus_uminus_real,comple1385675409528146559p_real(image_real_real(uminus_uminus_real,X5))) ).

% Inf_real_def
tff(fact_9866_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
tff(fact_9867_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_9868_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_ki(Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_9869_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,P2),Q2)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aTP_Lamm_kk(Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_9870_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_kl(D))) ).

% int_ge_less_than2_def
tff(fact_9871_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collec213857154873943460nt_int(produc4947309494688390418_int_o(aTP_Lamm_km(D))) ).

% int_ge_less_than_def
tff(fact_9872_uniformity__real__def,axiom,
    topolo1511823702728130853y_real = comple2936214249959783750l_real(image_2178119161166701260l_real(aTP_Lamm_ko,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_real_def
tff(fact_9873_uniformity__complex__def,axiom,
    topolo896644834953643431omplex = comple8358262395181532106omplex(image_5971271580939081552omplex(aTP_Lamm_kq,set_or5849166863359141190n_real(zero_zero_real))) ).

% uniformity_complex_def
tff(fact_9874_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_kr(F),G),at_top_nat)
          <=> bfun_nat_real(F,at_top_nat) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_9875_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Nb)
     => inj_on_real_real(aa_nat_fun_real_real(aTP_Lamm_id,Nb),top_top_set_real) ) ).

% inj_sgn_power
tff(fact_9876_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_9877_log__inj,axiom,
    ! [B2: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,one_one_real),B2)
     => inj_on_real_real(aa_rea7340605660300375871l_real(log,B2),set_or5849166863359141190n_real(zero_zero_real)) ) ).

% log_inj
tff(fact_9878_inj__Suc,axiom,
    ! [N: set_nat] : inj_on_nat_nat(suc,N) ).

% inj_Suc
tff(fact_9879_inj__on__diff__nat,axiom,
    ! [N: set_nat,K: nat] :
      ( ! [N2: nat] :
          ( member_nat(N2,N)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K),N2) )
     => inj_on_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hc,K),N) ) ).

% inj_on_diff_nat
tff(fact_9880_summable__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( 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_9881_suminf__reindex__mono,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( 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_9882_powr__real__of__int_H,axiom,
    ! [Xb: real,Nb: int] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( ( ( Xb != zero_zero_real )
          | aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),Nb) )
       => ( aa_real_real(aa_rea7340605660300375871l_real(powr_real,Xb),aa_int_real(ring_1_of_int_real,Nb)) = aa_int_real(power_int_real(Xb),Nb) ) ) ) ).

% powr_real_of_int'
tff(fact_9883_inj__on__char__of__nat,axiom,
    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_9884_suminf__reindex,axiom,
    ! [F: fun_nat_real,G: fun_nat_nat] :
      ( summable_real(F)
     => ( 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] :
                ( ~ 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_9885_pos__deriv__imp__strict__mono,axiom,
    ! [F: fun_real_real,F3: fun_real_real] :
      ( ! [X3: real] : has_fi5821293074295781190e_real(F,aa_real_real(F3,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(F3,X3))
       => order_7092887310737990675l_real(F) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_9886_pred__nat__def,axiom,
    pred_nat = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_ks)) ).

% pred_nat_def
tff(fact_9887_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I),J)
     => ( aa_nat_set_nat(set_or4665077453230672383an_nat(I),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = sup_sup_set_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(I),J),aa_nat_set_nat(set_or4665077453230672383an_nat(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_9888_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
tff(fact_9889_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
tff(fact_9890_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_9891_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_9892_tl__upt,axiom,
    ! [Mb: nat,Nb: nat] : tl_nat(upt(Mb,Nb)) = upt(aa_nat_nat(suc,Mb),Nb) ).

% tl_upt
tff(fact_9893_vanishes__mult__bounded,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( ? [A6: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),A6)
          & ! [N2: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N2))),A6) )
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kt,X5),Y6)) ) ) ).

% vanishes_mult_bounded
tff(fact_9894_vanishes__const,axiom,
    ! [C: rat] :
      ( vanishes(aTP_Lamm_ku(C))
    <=> ( C = zero_zero_rat ) ) ).

% vanishes_const
tff(fact_9895_vanishes__minus,axiom,
    ! [X5: fun_nat_rat] :
      ( vanishes(X5)
     => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kv,X5)) ) ).

% vanishes_minus
tff(fact_9896_vanishes__add,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( vanishes(X5)
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kw,X5),Y6)) ) ) ).

% vanishes_add
tff(fact_9897_vanishes__diff,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( vanishes(X5)
     => ( vanishes(Y6)
       => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6)) ) ) ).

% vanishes_diff
tff(fact_9898_vanishesD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( vanishes(X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [N5: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
           => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))),R2) ) ) ) ).

% vanishesD
tff(fact_9899_vanishesI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K4: nat] :
            ! [N2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N2)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N2))),R3) ) )
     => vanishes(X5) ) ).

% vanishesI
tff(fact_9900_vanishes__def,axiom,
    ! [X5: fun_nat_rat] :
      ( vanishes(X5)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N4))),R5) ) ) ) ).

% vanishes_def
tff(fact_9901_and__not__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( bit_and_not_num(Xb,Xaa) = Y )
     => ( accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(one) )
                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Y = none_num )
                     => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ( ( Xaa = one )
                     => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                       => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( Xb = aa_num_num(bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ( ( Xaa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,bit_and_not_num(M3,N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( Xb = aa_num_num(bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,bit_and_not_num(M3,N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_and_not_num_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_9902_and__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,Xb),Xaa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = some_num(one) )
               => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Y = none_num )
                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(one) )
                     => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ( ( Xaa = one )
                     => ( ( Y = none_num )
                       => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( Xb = aa_num_num(bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ( ( Xaa = one )
                           => ( ( Y = some_num(one) )
                             => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
                                 => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( Xb = aa_num_num(bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Y = case_o6005452278849405969um_num(some_num(one),aTP_Lamm_hg,aa_num_option_num(aa_num6390622462783740633on_num(bit_un7362597486090784418nd_num,M3),N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un4731106466462545111um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_9903_xor__num_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: option_num] :
      ( ( aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,Xb),Xaa) = Y )
     => ( accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = none_num )
               => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [N2: num] :
                  ( ( Xaa = aa_num_num(bit0,N2) )
                 => ( ( Y = some_num(aa_num_num(bit1,N2)) )
                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit0,N2))) ) ) )
           => ( ( ( Xb = one )
               => ! [N2: num] :
                    ( ( Xaa = aa_num_num(bit1,N2) )
                   => ( ( Y = some_num(aa_num_num(bit0,N2)) )
                     => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(one,aa_num_num(bit1,N2))) ) ) )
             => ( ! [M3: num] :
                    ( ( Xb = aa_num_num(bit0,M3) )
                   => ( ( Xaa = one )
                     => ( ( Y = some_num(aa_num_num(bit1,M3)) )
                       => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),one)) ) ) )
               => ( ! [M3: num] :
                      ( ( Xb = aa_num_num(bit0,M3) )
                     => ! [N2: num] :
                          ( ( Xaa = aa_num_num(bit0,N2) )
                         => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) )
                           => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit0,N2))) ) ) )
                 => ( ! [M3: num] :
                        ( ( Xb = aa_num_num(bit0,M3) )
                       => ! [N2: num] :
                            ( ( Xaa = aa_num_num(bit1,N2) )
                           => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit0,M3),aa_num_num(bit1,N2))) ) ) )
                   => ( ! [M3: num] :
                          ( ( Xb = aa_num_num(bit1,M3) )
                         => ( ( Xaa = one )
                           => ( ( Y = some_num(aa_num_num(bit0,M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),one)) ) ) )
                     => ( ! [M3: num] :
                            ( ( Xb = aa_num_num(bit1,M3) )
                           => ! [N2: num] :
                                ( ( Xaa = aa_num_num(bit0,N2) )
                               => ( ( Y = some_num(case_option_num_num(one,bit1,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2))) )
                                 => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit0,N2))) ) ) )
                       => ~ ! [M3: num] :
                              ( ( Xb = aa_num_num(bit1,M3) )
                             => ! [N2: num] :
                                  ( ( Xaa = aa_num_num(bit1,N2) )
                                 => ( ( Y = map_option_num_num(bit0,aa_num_option_num(aa_num6390622462783740633on_num(bit_un2480387367778600638or_num,M3),N2)) )
                                   => ~ accp_P3113834385874906142um_num(bit_un2901131394128224187um_rel,product_Pair_num_num(aa_num_num(bit1,M3),aa_num_num(bit1,N2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_9904_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( aa_num_num(bit_or_not_num_neg(Xb),Xaa) = Y )
     => ( accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(Xb,Xaa))
       => ( ( ( Xb = one )
           => ( ( Xaa = one )
             => ( ( Y = one )
               => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,one)) ) ) )
         => ( ( ( Xb = one )
             => ! [M3: num] :
                  ( ( Xaa = aa_num_num(bit0,M3) )
                 => ( ( Y = aa_num_num(bit1,M3) )
                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit0,M3))) ) ) )
           => ( ( ( Xb = one )
               => ! [M3: num] :
                    ( ( Xaa = aa_num_num(bit1,M3) )
                   => ( ( Y = aa_num_num(bit1,M3) )
                     => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(one,aa_num_num(bit1,M3))) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = aa_num_num(bit0,N2) )
                   => ( ( Xaa = one )
                     => ( ( Y = aa_num_num(bit0,one) )
                       => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),one)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = aa_num_num(bit0,N2) )
                     => ! [M3: num] :
                          ( ( Xaa = aa_num_num(bit0,M3) )
                         => ( ( Y = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) )
                           => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit0,M3))) ) ) )
                 => ( ! [N2: num] :
                        ( ( Xb = aa_num_num(bit0,N2) )
                       => ! [M3: num] :
                            ( ( Xaa = aa_num_num(bit1,M3) )
                           => ( ( Y = aa_num_num(bit0,aa_num_num(bit_or_not_num_neg(N2),M3)) )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit0,N2),aa_num_num(bit1,M3))) ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa_num_num(bit1,N2) )
                         => ( ( Xaa = one )
                           => ( ( Y = one )
                             => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),one)) ) ) )
                     => ( ! [N2: num] :
                            ( ( Xb = aa_num_num(bit1,N2) )
                           => ! [M3: num] :
                                ( ( Xaa = aa_num_num(bit0,M3) )
                               => ( ( Y = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) )
                                 => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit0,M3))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( Xb = aa_num_num(bit1,N2) )
                             => ! [M3: num] :
                                  ( ( Xaa = aa_num_num(bit1,M3) )
                                 => ( ( Y = aa_num_num(bitM,aa_num_num(bit_or_not_num_neg(N2),M3)) )
                                   => ~ accp_P3113834385874906142um_num(bit_or3848514188828904588eg_rel,product_Pair_num_num(aa_num_num(bit1,N2),aa_num_num(bit1,M3))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_9905_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_9906_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_9907_rcis__inverse,axiom,
    ! [R2: real,A: real] : aa_complex_complex(invers8013647133539491842omplex,aa_real_complex(rcis(R2),A)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),R2)),aa_real_real(uminus_uminus_real,A)) ).

% rcis_inverse
tff(fact_9908_Re__rcis,axiom,
    ! [R2: real,A: real] : aa_complex_real(re,aa_real_complex(rcis(R2),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(cos_real,A)) ).

% Re_rcis
tff(fact_9909_Im__rcis,axiom,
    ! [R2: real,A: real] : aa_complex_real(im,aa_real_complex(rcis(R2),A)) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R2),aa_real_real(sin_real,A)) ).

% Im_rcis
tff(fact_9910_cis__rcis__eq,axiom,
    ! [A: real] : aa_real_complex(cis,A) = aa_real_complex(rcis(one_one_real),A) ).

% cis_rcis_eq
tff(fact_9911_rcis__mult,axiom,
    ! [R1: real,A: real,R22: real,B2: real] : aa_complex_complex(times_times_complex(aa_real_complex(rcis(R1),A)),aa_real_complex(rcis(R22),B2)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(times_times_real,R1),R22)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,A),B2)) ).

% rcis_mult
tff(fact_9912_rcis__divide,axiom,
    ! [R1: real,A: real,R22: real,B2: real] : aa_complex_complex(divide1717551699836669952omplex(aa_real_complex(rcis(R1),A)),aa_real_complex(rcis(R22),B2)) = aa_real_complex(rcis(aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,R1),R22)),aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),B2)) ).

% rcis_divide
tff(fact_9913_rcis__def,axiom,
    ! [R2: real,A: real] : aa_real_complex(rcis(R2),A) = aa_complex_complex(times_times_complex(aa_real_complex(real_V4546457046886955230omplex,R2)),aa_real_complex(cis,A)) ).

% rcis_def
tff(fact_9914_DeMoivre2,axiom,
    ! [R2: real,A: real,Nb: nat] : aa_nat_complex(power_power_complex(aa_real_complex(rcis(R2),A)),Nb) = aa_real_complex(rcis(aa_nat_real(aa_real_fun_nat_real(power_power_real,R2),Nb)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(semiri5074537144036343181t_real,Nb)),A)) ).

% DeMoivre2
tff(fact_9915_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_ky,code_divmod_integer(K,numera6620942414471956472nteger(aa_num_num(bit0,one))))) ).

% nat_of_integer_code
tff(fact_9916_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_9917_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one_Code_integer) = one_one_nat ).

% nat_of_integer_code_post(2)
tff(fact_9918_cauchy__def,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
    <=> ! [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
         => ? [K3: nat] :
            ! [M6: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),M6)
             => ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M6)),aa_nat_rat(X5,N4)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_9919_cauchyI,axiom,
    ! [X5: fun_nat_rat] :
      ( ! [R3: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R3)
         => ? [K4: nat] :
            ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),M3)
             => ! [N2: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K4),N2)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M3)),aa_nat_rat(X5,N2)))),R3) ) ) )
     => cauchy(X5) ) ).

% cauchyI
tff(fact_9920_cauchy__inverse,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => cauchy(aTP_Lamm_kz(X5)) ) ) ).

% cauchy_inverse
tff(fact_9921_cauchy__diff,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6)) ) ) ).

% cauchy_diff
tff(fact_9922_cauchy__mult,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kt,X5),Y6)) ) ) ).

% cauchy_mult
tff(fact_9923_cauchy__const,axiom,
    ! [Xb: rat] : cauchy(aTP_Lamm_ku(Xb)) ).

% cauchy_const
tff(fact_9924_cauchy__minus,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => cauchy(aa_fun765758790203053448at_rat(aTP_Lamm_kv,X5)) ) ).

% cauchy_minus
tff(fact_9925_cauchy__add,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => cauchy(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kw,X5),Y6)) ) ) ).

% cauchy_add
tff(fact_9926_cauchy__imp__bounded,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ? [B4: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
          & ! [N5: nat] : aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))),B4) ) ) ).

% cauchy_imp_bounded
tff(fact_9927_vanishes__diff__inverse,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ( cauchy(Y6)
         => ( ~ vanishes(Y6)
           => ( vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6))
             => vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_la(X5),Y6)) ) ) ) ) ) ).

% vanishes_diff_inverse
tff(fact_9928_cauchy__not__vanishes__cases,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
            & ? [K2: nat] :
                ( ! [N5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(uminus_uminus_rat,aa_nat_rat(X5,N5))) )
                | ! [N5: nat] :
                    ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
                   => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_nat_rat(X5,N5)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_9929_cauchy__not__vanishes,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ vanishes(X5)
       => ? [B4: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),B4)
            & ? [K2: nat] :
              ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,B4),aa_rat_rat(abs_abs_rat,aa_nat_rat(X5,N5))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_9930_cauchyD,axiom,
    ! [X5: fun_nat_rat,R2: rat] :
      ( cauchy(X5)
     => ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R2)
       => ? [K2: nat] :
          ! [M4: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),M4)
           => ! [N5: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K2),N5)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,aa_rat_rat(abs_abs_rat,aa_rat_rat(minus_minus_rat(aa_nat_rat(X5,M4)),aa_nat_rat(X5,N5)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_9931_le__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y6))
        <=> ! [R5: rat] :
              ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
                 => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Y6,N4)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_9932_inverse__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,X5)) = $ite(vanishes(X5),zero_zero_real,aa_fun_nat_rat_real(real2,aTP_Lamm_kz(X5))) ) ) ).

% inverse_Real
tff(fact_9933_Real__induct,axiom,
    ! [P: fun_real_o,Xb: real] :
      ( ! [X7: fun_nat_rat] :
          ( cauchy(X7)
         => aa_real_o(P,aa_fun_nat_rat_real(real2,X7)) )
     => aa_real_o(P,Xb) ) ).

% Real_induct
tff(fact_9934_one__real__def,axiom,
    one_one_real = aa_fun_nat_rat_real(real2,aTP_Lamm_lb) ).

% one_real_def
tff(fact_9935_of__nat__Real,axiom,
    ! [Xb: nat] : aa_nat_real(semiri5074537144036343181t_real,Xb) = aa_fun_nat_rat_real(real2,aTP_Lamm_lc(Xb)) ).

% of_nat_Real
tff(fact_9936_zero__real__def,axiom,
    zero_zero_real = aa_fun_nat_rat_real(real2,aTP_Lamm_ld) ).

% zero_real_def
tff(fact_9937_of__int__Real,axiom,
    ! [Xb: int] : aa_int_real(ring_1_of_int_real,Xb) = aa_fun_nat_rat_real(real2,aTP_Lamm_le(Xb)) ).

% of_int_Real
tff(fact_9938_minus__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,X5)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_kv,X5)) ) ) ).

% minus_Real
tff(fact_9939_add__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kw,X5),Y6)) ) ) ) ).

% add_Real
tff(fact_9940_mult__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kt,X5),Y6)) ) ) ) ).

% mult_Real
tff(fact_9941_diff__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_fun_nat_rat_real(real2,X5)),aa_fun_nat_rat_real(real2,Y6)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6)) ) ) ) ).

% diff_Real
tff(fact_9942_eq__Real,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( ( aa_fun_nat_rat_real(real2,X5) = aa_fun_nat_rat_real(real2,Y6) )
        <=> vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6)) ) ) ) ).

% eq_Real
tff(fact_9943_not__positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( ~ aa_real_o(positive2,aa_fun_nat_rat_real(real2,X5))
      <=> ! [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
           => ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_nat_rat(X5,N4)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_9944_positive__Real,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => ( aa_real_o(positive2,aa_fun_nat_rat_real(real2,X5))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(X5,N4)) ) ) ) ) ).

% positive_Real
tff(fact_9945_Real_Opositive__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(positive2,Xb)
     => ( aa_real_o(positive2,Y)
       => aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Xb),Y)) ) ) ).

% Real.positive_mult
tff(fact_9946_Real_Opositive__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(positive2,Xb)
     => ( aa_real_o(positive2,Y)
       => aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Xb),Y)) ) ) ).

% Real.positive_add
tff(fact_9947_Real_Opositive__zero,axiom,
    ~ aa_real_o(positive2,zero_zero_real) ).

% Real.positive_zero
tff(fact_9948_Real_Opositive__minus,axiom,
    ! [Xb: real] :
      ( ~ aa_real_o(positive2,Xb)
     => ( ( Xb != zero_zero_real )
       => aa_real_o(positive2,aa_real_real(uminus_uminus_real,Xb)) ) ) ).

% Real.positive_minus
tff(fact_9949_less__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
    <=> aa_real_o(positive2,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Y),Xb)) ) ).

% less_real_def
tff(fact_9950_Real_Opositive_Orep__eq,axiom,
    ! [Xb: real] :
      ( aa_real_o(positive2,Xb)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(aa_real_fun_nat_rat(rep_real,Xb),N4)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_9951_inverse__real_Oabs__eq,axiom,
    ! [Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
     => ( aa_real_real(inverse_inverse_real,aa_fun_nat_rat_real(real2,Xb)) = aa_fun_nat_rat_real(real2,
            $ite(vanishes(Xb),aTP_Lamm_ld,aTP_Lamm_kz(Xb))) ) ) ).

% inverse_real.abs_eq
tff(fact_9952_realrel__refl,axiom,
    ! [X5: fun_nat_rat] :
      ( cauchy(X5)
     => aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,X5),X5) ) ).

% realrel_refl
tff(fact_9953_one__real_Orsp,axiom,
    aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,aTP_Lamm_lb),aTP_Lamm_lb) ).

% one_real.rsp
tff(fact_9954_zero__real_Orsp,axiom,
    aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,aTP_Lamm_ld),aTP_Lamm_ld) ).

% zero_real.rsp
tff(fact_9955_real_Oabs__induct,axiom,
    ! [P: fun_real_o,Xb: real] :
      ( ! [Y3: fun_nat_rat] :
          ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Y3),Y3)
         => aa_real_o(P,aa_fun_nat_rat_real(real2,Y3)) )
     => aa_real_o(P,Xb) ) ).

% real.abs_induct
tff(fact_9956_uminus__real_Oabs__eq,axiom,
    ! [Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
     => ( aa_real_real(uminus_uminus_real,aa_fun_nat_rat_real(real2,Xb)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aTP_Lamm_kv,Xb)) ) ) ).

% uminus_real.abs_eq
tff(fact_9957_plus__real_Oabs__eq,axiom,
    ! [Xaa: fun_nat_rat,Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xaa),Xaa)
     => ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
       => ( aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,aa_fun_nat_rat_real(real2,Xaa)),aa_fun_nat_rat_real(real2,Xb)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kw,Xaa),Xb)) ) ) ) ).

% plus_real.abs_eq
tff(fact_9958_times__real_Oabs__eq,axiom,
    ! [Xaa: fun_nat_rat,Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xaa),Xaa)
     => ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
       => ( aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_fun_nat_rat_real(real2,Xaa)),aa_fun_nat_rat_real(real2,Xb)) = aa_fun_nat_rat_real(real2,aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kt,Xaa),Xb)) ) ) ) ).

% times_real.abs_eq
tff(fact_9959_realrelI,axiom,
    ! [X5: fun_nat_rat,Y6: fun_nat_rat] :
      ( cauchy(X5)
     => ( cauchy(Y6)
       => ( vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X5),Y6))
         => aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,X5),Y6) ) ) ) ).

% realrelI
tff(fact_9960_realrel__def,axiom,
    ! [X4: fun_nat_rat,Xa: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,X4),Xa)
    <=> ( cauchy(X4)
        & cauchy(Xa)
        & vanishes(aa_fun765758790203053448at_rat(aTP_Lamm_kx(X4),Xa)) ) ) ).

% realrel_def
tff(fact_9961_Real_Opositive_Oabs__eq,axiom,
    ! [Xb: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,Xb),Xb)
     => ( aa_real_o(positive2,aa_fun_nat_rat_real(real2,Xb))
      <=> ? [R5: rat] :
            ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
            & ? [K3: nat] :
              ! [N4: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
               => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Xb,N4)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_9962_inverse__real__def,axiom,
    inverse_inverse_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_lf) ).

% inverse_real_def
tff(fact_9963_cr__real__def,axiom,
    ! [X4: fun_nat_rat,Xa: real] :
      ( aa_real_o(aa_fun8455130352500885026real_o(cr_real,X4),Xa)
    <=> ( aa_fun_nat_rat_o(aa_fun6961092367326905321_rat_o(realrel,X4),X4)
        & ( aa_fun_nat_rat_real(real2,X4) = Xa ) ) ) ).

% cr_real_def
tff(fact_9964_uminus__real__def,axiom,
    uminus_uminus_real = aa_fun8686829112900759002l_real(map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_kv) ).

% uminus_real_def
tff(fact_9965_times__real__def,axiom,
    times_times_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_kt) ).

% times_real_def
tff(fact_9966_plus__real__def,axiom,
    plus_plus_real = map_fu2502140041903272327l_real(rep_real,map_fu3926683916856352828t_real(rep_real,real2),aTP_Lamm_kw) ).

% plus_real_def
tff(fact_9967_Real_Opositive_Orsp,axiom,
    bNF_re7504242029904815401at_o_o(realrel,fequal_o,aTP_Lamm_lg,aTP_Lamm_lg) ).

% Real.positive.rsp
tff(fact_9968_plus__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_kw,aTP_Lamm_kw) ).

% plus_real.rsp
tff(fact_9969_uminus__real_Orsp,axiom,
    aa_fun175815717583936595_rat_o(aa_fun9037873928781546661_rat_o(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_kv),aTP_Lamm_kv) ).

% uminus_real.rsp
tff(fact_9970_times__real_Orsp,axiom,
    bNF_re769765379126235449at_rat(realrel,bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_kt,aTP_Lamm_kt) ).

% times_real.rsp
tff(fact_9971_sub_Orsp,axiom,
    bNF_re6378542964492772123um_int(fequal_num,bNF_re1822329894187522285nt_int(fequal_num,fequal_int),aTP_Lamm_lh,aTP_Lamm_lh) ).

% sub.rsp
tff(fact_9972_times__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),times_times_nat,times_times_nat) ).

% times_natural.rsp
tff(fact_9973_times__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),times_times_int,times_times_int) ).

% times_integer.rsp
tff(fact_9974_minus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),minus_minus_int,minus_minus_int) ).

% minus_integer.rsp
tff(fact_9975_minus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),minus_minus_nat,minus_minus_nat) ).

% minus_natural.rsp
tff(fact_9976_divide__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),divide_divide_nat,divide_divide_nat) ).

% divide_natural.rsp
tff(fact_9977_divide__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),divide_divide_int,divide_divide_int) ).

% divide_integer.rsp
tff(fact_9978_modulo__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),modulo_modulo_nat,modulo_modulo_nat) ).

% modulo_natural.rsp
tff(fact_9979_modulo__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),modulo_modulo_int,modulo_modulo_int) ).

% modulo_integer.rsp
tff(fact_9980_Suc_Orsp,axiom,
    aa_fun_nat_nat_o(aa_fun2341495900663639033_nat_o(bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),suc),suc) ).

% Suc.rsp
tff(fact_9981_dup_Orsp,axiom,
    aa_fun_int_int_o(aa_fun2545150047610057353_int_o(bNF_re4712519889275205905nt_int(fequal_int,fequal_int),aTP_Lamm_li),aTP_Lamm_li) ).

% dup.rsp
tff(fact_9982_plus__natural_Orsp,axiom,
    bNF_re5162689864697376059at_nat(fequal_nat,bNF_re5653821019739307937at_nat(fequal_nat,fequal_nat),plus_plus_nat,plus_plus_nat) ).

% plus_natural.rsp
tff(fact_9983_plus__integer_Orsp,axiom,
    bNF_re5717476210804479843nt_int(fequal_int,bNF_re4712519889275205905nt_int(fequal_int,fequal_int),plus_plus_int,plus_plus_int) ).

% plus_integer.rsp
tff(fact_9984_less__integer_Orsp,axiom,
    bNF_re8654322613993568759_int_o(fequal_int,bNF_re5089333283451836215nt_o_o(fequal_int,fequal_o),ord_less_int,ord_less_int) ).

% less_integer.rsp
tff(fact_9985_less__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_nat,ord_less_nat) ).

% less_natural.rsp
tff(fact_9986_less__eq__natural_Orsp,axiom,
    bNF_re1448341924010609215_nat_o(fequal_nat,bNF_re4705727531993890431at_o_o(fequal_nat,fequal_o),ord_less_eq_nat,ord_less_eq_nat) ).

% less_eq_natural.rsp
tff(fact_9987_le__enumerate,axiom,
    ! [S2: set_nat,Nb: nat] :
      ( ~ finite_finite_nat(S2)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S2),Nb)) ) ).

% le_enumerate
tff(fact_9988_inverse__real_Orsp,axiom,
    aa_fun175815717583936595_rat_o(aa_fun9037873928781546661_rat_o(bNF_re9178625837396562677at_rat(realrel,realrel),aTP_Lamm_lf),aTP_Lamm_lf) ).

% inverse_real.rsp
tff(fact_9989_finite__le__enumerate,axiom,
    ! [S2: set_nat,Nb: nat] :
      ( finite_finite_nat(S2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Nb),finite_card_nat(S2))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Nb),aa_nat_nat(infini8530281810654367211te_nat(S2),Nb)) ) ) ).

% finite_le_enumerate
tff(fact_9990_Real_Opositive_Otransfer,axiom,
    aa_fun_real_o_o(aa_fun7909102309695209382al_o_o(bNF_re3218747500659316912al_o_o(pcr_real,fequal_o),aTP_Lamm_lg),positive2) ).

% Real.positive.transfer
tff(fact_9991_plus__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_lj,plus_plus_rat) ).

% plus_rat.transfer
tff(fact_9992_real_Orel__eq__transfer,axiom,
    bNF_re6090955117988708457real_o(pcr_real,bNF_re3218747500659316912al_o_o(pcr_real,fequal_o),realrel,fequal_real) ).

% real.rel_eq_transfer
tff(fact_9993_real_Opcr__cr__eq,axiom,
    pcr_real = cr_real ).

% real.pcr_cr_eq
tff(fact_9994_zero__real_Otransfer,axiom,
    aa_real_o(aa_fun8455130352500885026real_o(pcr_real,aTP_Lamm_ld),zero_zero_real) ).

% zero_real.transfer
tff(fact_9995_one__real_Otransfer,axiom,
    aa_real_o(aa_fun8455130352500885026real_o(pcr_real,aTP_Lamm_lb),one_one_real) ).

% one_real.transfer
tff(fact_9996_cr__real__eq,axiom,
    ! [X4: fun_nat_rat,Xa: real] :
      ( aa_real_o(aa_fun8455130352500885026real_o(pcr_real,X4),Xa)
    <=> ( cauchy(X4)
        & ( aa_fun_nat_rat_real(real2,X4) = Xa ) ) ) ).

% cr_real_eq
tff(fact_9997_uminus__real_Otransfer,axiom,
    aa_fun_real_real_o(aa_fun1584707981518017559real_o(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_kv),uminus_uminus_real) ).

% uminus_real.transfer
tff(fact_9998_plus__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_kw,plus_plus_real) ).

% plus_real.transfer
tff(fact_9999_times__real_Otransfer,axiom,
    bNF_re1162444993429354318l_real(pcr_real,bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_kt,times_times_real) ).

% times_real.transfer
tff(fact_10000_inverse__real_Otransfer,axiom,
    aa_fun_real_real_o(aa_fun1584707981518017559real_o(bNF_re237660433471117187t_real(pcr_real,pcr_real),aTP_Lamm_lf),inverse_inverse_real) ).

% inverse_real.transfer
tff(fact_10001_times__rat_Otransfer,axiom,
    bNF_re6323452265856231590at_rat(pcr_rat,bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat),aTP_Lamm_lk,times_times_rat) ).

% times_rat.transfer
tff(fact_10002_Rat_Opositive_Otransfer,axiom,
    bNF_re1494630372529172596at_o_o(pcr_rat,fequal_o,aTP_Lamm_ll,positive) ).

% Rat.positive.transfer
tff(fact_10003_times__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_gr),times_times_int) ).

% times_int.transfer
tff(fact_10004_Least__eq__0,axiom,
    ! [P: fun_nat_o] :
      ( aa_nat_o(P,zero_zero_nat)
     => ( ord_Least_nat(P) = zero_zero_nat ) ) ).

% Least_eq_0
tff(fact_10005_Least__Suc,axiom,
    ! [P: fun_nat_o,Nb: nat] :
      ( aa_nat_o(P,Nb)
     => ( ~ aa_nat_o(P,zero_zero_nat)
       => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(aTP_Lamm_jr(P))) ) ) ) ).

% Least_Suc
tff(fact_10006_Least__Suc2,axiom,
    ! [P: fun_nat_o,Nb: nat,Q: fun_nat_o,Mb: nat] :
      ( aa_nat_o(P,Nb)
     => ( aa_nat_o(Q,Mb)
       => ( ~ aa_nat_o(P,zero_zero_nat)
         => ( ! [K2: nat] :
                ( aa_nat_o(P,aa_nat_nat(suc,K2))
              <=> aa_nat_o(Q,K2) )
           => ( ord_Least_nat(P) = aa_nat_nat(suc,ord_Least_nat(Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_10007_Sup__real__def,axiom,
    ! [X5: set_real] : comple1385675409528146559p_real(X5) = ord_Least_real(aTP_Lamm_lm(X5)) ).

% Sup_real_def
tff(fact_10008_nat_Otransfer,axiom,
    bNF_re4555766996558763186at_nat(pcr_int,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),nat2) ).

% nat.transfer
tff(fact_10009_one__int_Otransfer,axiom,
    aa_int_o(aa_Pro2912763917461432128_int_o(pcr_int,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_10010_less__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_gt),ord_less_int) ).

% less_int.transfer
tff(fact_10011_less__eq__int_Otransfer,axiom,
    bNF_re4195927319312321177_int_o(pcr_int,bNF_re6644619430987730960nt_o_o(pcr_int,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_gv),ord_less_eq_int) ).

% less_eq_int.transfer
tff(fact_10012_plus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_gx),plus_plus_int) ).

% plus_int.transfer
tff(fact_10013_minus__int_Otransfer,axiom,
    bNF_re8869769828555196650nt_int(pcr_int,bNF_re7400052026677387805at_int(pcr_int,pcr_int),produc4528391901023682385at_nat(aTP_Lamm_gz),minus_minus_int) ).

% minus_int.transfer
tff(fact_10014_of__rat__Real,axiom,
    ! [Xb: rat] : aa_rat_real(field_7254667332652039916t_real,Xb) = aa_fun_nat_rat_real(real2,aTP_Lamm_ku(Xb)) ).

% of_rat_Real
tff(fact_10015_of__rat__dense,axiom,
    ! [Xb: real,Y: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),Y)
     => ? [Q3: rat] :
          ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_rat_real(field_7254667332652039916t_real,Q3))
          & aa_real_o(aa_real_fun_real_o(ord_less_real,aa_rat_real(field_7254667332652039916t_real,Q3)),Y) ) ) ).

% of_rat_dense
tff(fact_10016_less__RealD,axiom,
    ! [Y6: fun_nat_rat,Xb: real] :
      ( cauchy(Y6)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_fun_nat_rat_real(real2,Y6))
       => ? [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_real,Xb),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2))) ) ) ).

% less_RealD
tff(fact_10017_le__RealI,axiom,
    ! [Y6: fun_nat_rat,Xb: real] :
      ( cauchy(Y6)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(Y6,N2)))
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),aa_fun_nat_rat_real(real2,Y6)) ) ) ).

% le_RealI
tff(fact_10018_Real__leI,axiom,
    ! [X5: fun_nat_rat,Y: real] :
      ( cauchy(X5)
     => ( ! [N2: nat] : aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_rat_real(field_7254667332652039916t_real,aa_nat_rat(X5,N2))),Y)
       => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_fun_nat_rat_real(real2,X5)),Y) ) ) ).

% Real_leI
tff(fact_10019_times__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_gr),produc4528391901023682385at_nat(aTP_Lamm_gr)) ).

% times_int.rsp
tff(fact_10020_intrel__iff,axiom,
    ! [Xb: nat,Y: nat,U: nat,V: nat] :
      ( aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(U),V))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Xb),V) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,U),Y) ) ) ).

% intrel_iff
tff(fact_10021_nat_Orsp,axiom,
    bNF_re8246922863344978751at_nat(intrel,fequal_nat,produc6842872674320459806at_nat(minus_minus_nat),produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat.rsp
tff(fact_10022_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(aa_Pro989595756709598047_nat_o(intrel,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(one_one_nat),zero_zero_nat)) ).

% one_int.rsp
tff(fact_10023_intrel__def,axiom,
    intrel = produc6395135316500328296_nat_o(aTP_Lamm_lo) ).

% intrel_def
tff(fact_10024_less__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_gt),produc6395135316500328296_nat_o(aTP_Lamm_gt)) ).

% less_int.rsp
tff(fact_10025_less__eq__int_Orsp,axiom,
    bNF_re4103222659376994899_nat_o(intrel,bNF_re3666534408544137501at_o_o(intrel,fequal_o),produc6395135316500328296_nat_o(aTP_Lamm_gv),produc6395135316500328296_nat_o(aTP_Lamm_gv)) ).

% less_eq_int.rsp
tff(fact_10026_plus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_gx),produc4528391901023682385at_nat(aTP_Lamm_gx)) ).

% plus_int.rsp
tff(fact_10027_minus__int_Orsp,axiom,
    bNF_re6214786843070381989at_nat(intrel,bNF_re2241393799969408733at_nat(intrel,intrel),produc4528391901023682385at_nat(aTP_Lamm_gz),produc4528391901023682385at_nat(aTP_Lamm_gz)) ).

% minus_int.rsp
tff(fact_10028_has__vector__derivative__id,axiom,
    ! [Net: filter_real] : has_ve631408500373753343e_real(aTP_Lamm_az,one_one_real,Net) ).

% has_vector_derivative_id
tff(fact_10029_has__field__derivative__iff__has__vector__derivative,axiom,
    ! [F: fun_real_real,Y: real,F4: filter_real] :
      ( has_fi5821293074295781190e_real(F,Y,F4)
    <=> has_ve631408500373753343e_real(F,Y,F4) ) ).

% has_field_derivative_iff_has_vector_derivative
tff(fact_10030_plus__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_lj,aTP_Lamm_lj) ).

% plus_rat.rsp
tff(fact_10031_ratrel__iff,axiom,
    ! [Xb: product_prod_int_int,Y: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Y)
    <=> ( ( product_snd_int_int(Xb) != zero_zero_int )
        & ( product_snd_int_int(Y) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Y)),product_snd_int_int(Xb)) ) ) ) ).

% ratrel_iff
tff(fact_10032_ratrel__def,axiom,
    ! [X4: product_prod_int_int,Xa: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X4),Xa)
    <=> ( ( product_snd_int_int(X4) != zero_zero_int )
        & ( product_snd_int_int(Xa) != zero_zero_int )
        & ( aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(X4)),product_snd_int_int(Xa)) = aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xa)),product_snd_int_int(X4)) ) ) ) ).

% ratrel_def
tff(fact_10033_times__rat_Orsp,axiom,
    bNF_re4734134812813710757nt_int(ratrel,bNF_re7145576690424134365nt_int(ratrel,ratrel),aTP_Lamm_lk,aTP_Lamm_lk) ).

% times_rat.rsp
tff(fact_10034_Rat_Opositive_Orsp,axiom,
    bNF_re8699439704749558557nt_o_o(ratrel,fequal_o,aTP_Lamm_ll,aTP_Lamm_ll) ).

% Rat.positive.rsp
tff(fact_10035_plus__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xaa),Xaa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xaa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xaa)),product_snd_int_int(Xb))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Xaa))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xaa)),product_snd_int_int(Xb)))) ) ) ) ).

% plus_rat.abs_eq
tff(fact_10036_Rat_Opositive_Oabs__eq,axiom,
    ! [Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
     => ( aa_rat_o(positive,aa_Pro6166618844887768661nt_rat(abs_Rat,Xb))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xb)),product_snd_int_int(Xb))) ) ) ).

% Rat.positive.abs_eq
tff(fact_10037_times__rat_Oabs__eq,axiom,
    ! [Xaa: product_prod_int_int,Xb: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xaa),Xaa)
     => ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Xb),Xb)
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,Xaa)),aa_Pro6166618844887768661nt_rat(abs_Rat,Xb)) = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Xaa)),product_fst_int_int(Xb)),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Xaa)),product_snd_int_int(Xb)))) ) ) ) ).

% times_rat.abs_eq
tff(fact_10038_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I),J)
     => ( last_nat(upt(I,J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_10039_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_du(F),M5),at_top_nat)
     => ( ! [M3: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),M3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,N2)),aa_nat_real(F,M3)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_10040_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: fun_nat_real,M5: nat] :
      ( bfun_nat_real(aa_nat_fun_nat_real(aTP_Lamm_du(F),M5),at_top_nat)
     => ( ! [M3: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M5),M3)
           => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
             => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(F,M3)),aa_nat_real(F,N2)) ) )
       => topolo7531315842566124627t_real(F) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_10041_Bseq__mono__convergent,axiom,
    ! [X5: fun_nat_real] :
      ( bfun_nat_real(X5,at_top_nat)
     => ( ! [M3: nat,N2: nat] :
            ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M3),N2)
           => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_nat_real(X5,M3)),aa_nat_real(X5,N2)) )
       => topolo7531315842566124627t_real(X5) ) ) ).

% Bseq_mono_convergent
tff(fact_10042_convergent__realpow,axiom,
    ! [Xb: real] :
      ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Xb)
     => ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Xb),one_one_real)
       => topolo7531315842566124627t_real(aa_real_fun_nat_real(power_power_real,Xb)) ) ) ).

% convergent_realpow
tff(fact_10043_filtermap__at__right__shift,axiom,
    ! [D: real,A: real] : filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lp,D),topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A))) = topolo2177554685111907308n_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),D),set_or5849166863359141190n_real(aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,A),D))) ).

% filtermap_at_right_shift
tff(fact_10044_at__right__to__0,axiom,
    ! [A: real] : topolo2177554685111907308n_real(A,set_or5849166863359141190n_real(A)) = filtermap_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lq,A),topolo2177554685111907308n_real(zero_zero_real,set_or5849166863359141190n_real(zero_zero_real))) ).

% at_right_to_0
tff(fact_10045_pair__lessI2,axiom,
    ! [A: nat,B2: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,S),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_10046_pair__less__iff1,axiom,
    ! [Xb: nat,Y: nat,Zb: nat] :
      ( member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Zb)),fun_pair_less)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Zb) ) ).

% pair_less_iff1
tff(fact_10047_pair__lessI1,axiom,
    ! [A: nat,B2: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_10048_pair__leqI2,axiom,
    ! [A: nat,B2: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,S),Ta)
       => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_10049_pair__leqI1,axiom,
    ! [A: nat,B2: nat,S: nat,Ta: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B2)
     => member8206827879206165904at_nat(produc6161850002892822231at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(A),S),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(B2),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_10050_bot__nat__0_Oordering__top__axioms,axiom,
    aa_nat_o(ordering_top_nat(aTP_Lamm_eh,aTP_Lamm_ei),zero_zero_nat) ).

% bot_nat_0.ordering_top_axioms
tff(fact_10051_set__encode__vimage__Suc,axiom,
    ! [A3: set_nat] : aa_set_nat_nat(nat_set_encode,vimage_nat_nat(suc,A3)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_set_nat_nat(nat_set_encode,A3)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% set_encode_vimage_Suc
tff(fact_10052_euclidean__size__int__def,axiom,
    euclid4774559944035922753ze_int = comp_int_nat_int(nat2,abs_abs_int) ).

% euclidean_size_int_def
tff(fact_10053_finite__vimage__Suc__iff,axiom,
    ! [F4: set_nat] :
      ( finite_finite_nat(vimage_nat_nat(suc,F4))
    <=> finite_finite_nat(F4) ) ).

% finite_vimage_Suc_iff
tff(fact_10054_vimage__Suc__insert__Suc,axiom,
    ! [Nb: nat,A3: set_nat] : vimage_nat_nat(suc,insert_nat(aa_nat_nat(suc,Nb),A3)) = insert_nat(Nb,vimage_nat_nat(suc,A3)) ).

% vimage_Suc_insert_Suc
tff(fact_10055_vimage__Suc__insert__0,axiom,
    ! [A3: set_nat] : vimage_nat_nat(suc,insert_nat(zero_zero_nat,A3)) = vimage_nat_nat(suc,A3) ).

% vimage_Suc_insert_0
tff(fact_10056_set__decode__div__2,axiom,
    ! [Xb: nat] : aa_nat_set_nat(nat_set_decode,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Xb),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = vimage_nat_nat(suc,aa_nat_set_nat(nat_set_decode,Xb)) ).

% set_decode_div_2
tff(fact_10057_abs__division__segment,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ).

% abs_division_segment
tff(fact_10058_division__segment__nat__def,axiom,
    ! [Nb: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,Nb) = one_one_nat ).

% division_segment_nat_def
tff(fact_10059_division__segment__eq__sgn,axiom,
    ! [K: int] :
      ( ( K != zero_zero_int )
     => ( aa_int_int(euclid3395696857347342551nt_int,K) = aa_int_int(sgn_sgn_int,K) ) ) ).

% division_segment_eq_sgn
tff(fact_10060_division__segment__int__def,axiom,
    ! [K: int] :
      aa_int_int(euclid3395696857347342551nt_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% division_segment_int_def
tff(fact_10061_less__eq__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Mb),Nb)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_lr(Mb),$true),Nb) ) ).

% less_eq_enat_def
tff(fact_10062_transp__realrel,axiom,
    transp_fun_nat_rat(realrel) ).

% transp_realrel
tff(fact_10063_less__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Mb),Nb)
    <=> aa_Extended_enat_o(extended_case_enat_o(aTP_Lamm_ls(Nb),$false),Mb) ) ).

% less_enat_def
tff(fact_10064_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod_int_int] : adjust_div(Qr) = produc8211389475949308722nt_int(aTP_Lamm_lt,Qr) ).

% Divides.adjust_div_def
tff(fact_10065_rat__floor__code,axiom,
    ! [P2: rat] : aa_rat_int(archim3151403230148437115or_rat,P2) = produc8211389475949308722nt_int(divide_divide_int,quotient_of(P2)) ).

% rat_floor_code
tff(fact_10066_prod__decode__triangle__add,axiom,
    ! [K: nat,Mb: nat] : nat_prod_decode(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,K)),Mb)) = nat_prod_decode_aux(K,Mb) ).

% prod_decode_triangle_add
tff(fact_10067_list__decode_Opinduct,axiom,
    ! [A0: nat,P: fun_nat_o] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),A0)
     => ( ( aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat)
         => aa_nat_o(P,zero_zero_nat) )
       => ( ! [N2: nat] :
              ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2))
             => ( ! [X4: nat,Y4: nat] :
                    ( ( aa_nat7603221163087113075at_nat(product_Pair_nat_nat(X4),Y4) = nat_prod_decode(N2) )
                   => aa_nat_o(P,Y4) )
               => aa_nat_o(P,aa_nat_nat(suc,N2)) ) )
         => aa_nat_o(P,A0) ) ) ) ).

% list_decode.pinduct
tff(fact_10068_list__decode_Oelims,axiom,
    ! [Xb: nat,Y: list_nat] :
      ( ( nat_list_decode(Xb) = Y )
     => ( ( ( Xb = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( Xb = aa_nat_nat(suc,N2) )
             => ( Y != produc2761476792215241774st_nat(aTP_Lamm_lu,nat_prod_decode(N2)) ) ) ) ) ).

% list_decode.elims
tff(fact_10069_list__decode_Opsimps_I2_J,axiom,
    ! [Nb: nat] :
      ( aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,Nb))
     => ( nat_list_decode(aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_lu,nat_prod_decode(Nb)) ) ) ).

% list_decode.psimps(2)
tff(fact_10070_list__decode_Osimps_I2_J,axiom,
    ! [Nb: nat] : nat_list_decode(aa_nat_nat(suc,Nb)) = produc2761476792215241774st_nat(aTP_Lamm_lu,nat_prod_decode(Nb)) ).

% list_decode.simps(2)
tff(fact_10071_list__decode_Opelims,axiom,
    ! [Xb: nat,Y: list_nat] :
      ( ( nat_list_decode(Xb) = Y )
     => ( aa_nat_o(accp_nat(nat_list_decode_rel),Xb)
       => ( ( ( Xb = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ aa_nat_o(accp_nat(nat_list_decode_rel),zero_zero_nat) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa_nat_nat(suc,N2) )
               => ( ( Y = produc2761476792215241774st_nat(aTP_Lamm_lu,nat_prod_decode(N2)) )
                 => ~ aa_nat_o(accp_nat(nat_list_decode_rel),aa_nat_nat(suc,N2)) ) ) ) ) ) ).

% list_decode.pelims
tff(fact_10072_compute__powr__real,axiom,
    ! [B2: real,I: real] :
      aa_real_real(aa_rea7340605660300375871l_real(powr_real2,B2),I) = $ite(
        aa_real_o(aa_real_fun_real_o(ord_less_eq_real,B2),zero_zero_real),
        abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_lv(B2),I)),
        $ite(
          aa_int_real(ring_1_of_int_real,aa_real_int(archim6058952711729229775r_real,I)) = I,
          $ite(aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),I),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,I))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(aa_real_fun_nat_real(power_power_real,B2),aa_int_nat(nat2,aa_real_int(archim6058952711729229775r_real,aa_real_real(uminus_uminus_real,I)))))),
          abort_real(literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero_literal))))))))))))))))))))))))))))))))))),aa_rea7683167438103758690t_real(aTP_Lamm_lv(B2),I)) ) ) ).

% compute_powr_real
tff(fact_10073_pairs__le__eq__Sigma,axiom,
    ! [Mb: nat] : collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_lw(Mb))) = produc457027306803732586at_nat(aa_nat_set_nat(set_ord_atMost_nat,Mb),aTP_Lamm_lx(Mb)) ).

% pairs_le_eq_Sigma
tff(fact_10074_product__atMost__eq__Un,axiom,
    ! [A3: set_nat,Mb: nat] : produc457027306803732586at_nat(A3,aTP_Lamm_ly(Mb)) = sup_su6327502436637775413at_nat(produc457027306803732586at_nat(A3,aTP_Lamm_lx(Mb)),produc457027306803732586at_nat(A3,aTP_Lamm_lz(Mb))) ).

% product_atMost_eq_Un
tff(fact_10075_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
tff(fact_10076_Real_Opositive__def,axiom,
    positive2 = map_fu6719675619867407465at_o_o(rep_real,id_o,aTP_Lamm_lg) ).

% Real.positive_def
tff(fact_10077_euclidean__size__nat__def,axiom,
    euclid4777050414544973029ze_nat = id_nat ).

% euclidean_size_nat_def
tff(fact_10078_times__rat__def,axiom,
    times_times_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_lk) ).

% times_rat_def
tff(fact_10079_plus__rat__def,axiom,
    plus_plus_rat = map_fu5968964422797239547at_rat(rep_Rat,map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat),aTP_Lamm_lj) ).

% plus_rat_def
tff(fact_10080_Rat_Opositive__def,axiom,
    positive = map_fu898904425404107465nt_o_o(rep_Rat,id_o,aTP_Lamm_ll) ).

% Rat.positive_def
tff(fact_10081_nat__def,axiom,
    nat2 = map_fu2345160673673942751at_nat(rep_Integ,id_nat,produc6842872674320459806at_nat(minus_minus_nat)) ).

% nat_def
tff(fact_10082_less__int__def,axiom,
    ord_less_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_gt)) ).

% less_int_def
tff(fact_10083_less__eq__int__def,axiom,
    ord_less_eq_int = map_fu3191548434056052140_int_o(rep_Integ,map_fu4826362097070443709at_o_o(rep_Integ,id_o),produc6395135316500328296_nat_o(aTP_Lamm_gv)) ).

% less_eq_int_def
tff(fact_10084_plus__int__def,axiom,
    plus_plus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_gx)) ).

% plus_int_def
tff(fact_10085_minus__int__def,axiom,
    minus_minus_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_gz)) ).

% minus_int_def
tff(fact_10086_times__int__def,axiom,
    times_times_int = map_fu8294830438409148311nt_int(rep_Integ,map_fu3667384564859982768at_int(rep_Integ,abs_Integ),produc4528391901023682385at_nat(aTP_Lamm_gr)) ).

% times_int_def
tff(fact_10087_eventually__prod__sequentially,axiom,
    ! [P: fun_Pr4796943543331071514_nat_o] :
      ( eventu1038000079068216329at_nat(P,prod_filter_nat_nat(at_top_nat,at_top_nat))
    <=> ? [N7: nat] :
        ! [M6: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),M6)
         => ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N7),N4)
             => aa_Pro6150637604445964947_nat_o(P,aa_nat7603221163087113075at_nat(product_Pair_nat_nat(N4),M6)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_10088_less__eq,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi6264000038957366511cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% less_eq
tff(fact_10089_numeral__le__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(numera1916890842035813515d_enat(Mb)),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_le_enat_iff
tff(fact_10090_enat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( extended_enat2(Nat) = extended_enat2(Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% enat.inject
tff(fact_10091_enat__ord__simps_I2_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(extended_enat2(Mb)),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Mb),Nb) ) ).

% enat_ord_simps(2)
tff(fact_10092_enat__ord__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(extended_enat2(Mb)),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% enat_ord_simps(1)
tff(fact_10093_plus__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : plus_p3455044024723400733d_enat(extended_enat2(Mb),extended_enat2(Nb)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Mb),Nb)) ).

% plus_enat_simps(1)
tff(fact_10094_idiff__enat__0__right,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(Nb,extended_enat2(zero_zero_nat)) = Nb ).

% idiff_enat_0_right
tff(fact_10095_idiff__enat__0,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(extended_enat2(zero_zero_nat),Nb) = extended_enat2(zero_zero_nat) ).

% idiff_enat_0
tff(fact_10096_idiff__enat__enat,axiom,
    ! [A: nat,B2: nat] : minus_3235023915231533773d_enat(extended_enat2(A),extended_enat2(B2)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B2)) ).

% idiff_enat_enat
tff(fact_10097_times__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : times_7803423173614009249d_enat(extended_enat2(Mb),extended_enat2(Nb)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb)) ).

% times_enat_simps(1)
tff(fact_10098_max__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Mb),Nb)) ).

% max_enat_simps(1)
tff(fact_10099_min__enat__simps_I1_J,axiom,
    ! [Mb: nat,Nb: nat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,extended_enat2(Mb)),extended_enat2(Nb)) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Mb),Nb)) ).

% min_enat_simps(1)
tff(fact_10100_numeral__less__enat__iff,axiom,
    ! [Mb: num,Nb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(numera1916890842035813515d_enat(Mb)),extended_enat2(Nb))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_num_nat(numeral_numeral_nat,Mb)),Nb) ) ).

% numeral_less_enat_iff
tff(fact_10101_zero__enat__def,axiom,
    zero_z5237406670263579293d_enat = extended_enat2(zero_zero_nat) ).

% zero_enat_def
tff(fact_10102_enat__0__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = zero_z5237406670263579293d_enat )
    <=> ( Xb = zero_zero_nat ) ) ).

% enat_0_iff(1)
tff(fact_10103_enat__0__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( zero_z5237406670263579293d_enat = extended_enat2(Xb) )
    <=> ( Xb = zero_zero_nat ) ) ).

% enat_0_iff(2)
tff(fact_10104_of__nat__eq__enat,axiom,
    ! [Nb: nat] : semiri4216267220026989637d_enat(Nb) = extended_enat2(Nb) ).

% of_nat_eq_enat
tff(fact_10105_enat__iless,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),extended_enat2(Mb))
     => ? [K2: nat] : Nb = extended_enat2(K2) ) ).

% enat_iless
tff(fact_10106_less__enatE,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),extended_enat2(Mb))
     => ~ ! [K2: nat] :
            ( ( Nb = extended_enat2(K2) )
           => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,K2),Mb) ) ) ).

% less_enatE
tff(fact_10107_one__enat__def,axiom,
    one_on7984719198319812577d_enat = extended_enat2(one_one_nat) ).

% one_enat_def
tff(fact_10108_enat__1__iff_I1_J,axiom,
    ! [Xb: nat] :
      ( ( extended_enat2(Xb) = one_on7984719198319812577d_enat )
    <=> ( Xb = one_one_nat ) ) ).

% enat_1_iff(1)
tff(fact_10109_enat__1__iff_I2_J,axiom,
    ! [Xb: nat] :
      ( ( one_on7984719198319812577d_enat = extended_enat2(Xb) )
    <=> ( Xb = one_one_nat ) ) ).

% enat_1_iff(2)
tff(fact_10110_numeral__eq__enat,axiom,
    ! [K: num] : numera1916890842035813515d_enat(K) = extended_enat2(aa_num_nat(numeral_numeral_nat,K)) ).

% numeral_eq_enat
tff(fact_10111_Suc__ile__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(extended_enat2(aa_nat_nat(suc,Mb))),Nb)
    <=> aa_Extended_enat_o(ord_le72135733267957522d_enat(extended_enat2(Mb)),Nb) ) ).

% Suc_ile_eq
tff(fact_10112_finite__enat__bounded,axiom,
    ! [A3: set_Extended_enat,Nb: nat] :
      ( ! [Y3: extended_enat] :
          ( member_Extended_enat(Y3,A3)
         => aa_Extended_enat_o(ord_le2932123472753598470d_enat(Y3),extended_enat2(Nb)) )
     => finite4001608067531595151d_enat(A3) ) ).

% finite_enat_bounded
tff(fact_10113_enat__ile,axiom,
    ! [Nb: extended_enat,Mb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),extended_enat2(Mb))
     => ? [K2: nat] : Nb = extended_enat2(K2) ) ).

% enat_ile
tff(fact_10114_pred__nat__trancl__eq__le,axiom,
    ! [Mb: nat,Nb: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Mb),Nb),transi2905341329935302413cl_nat(pred_nat))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Mb),Nb) ) ).

% pred_nat_trancl_eq_le
tff(fact_10115_iadd__le__enat__iff,axiom,
    ! [Xb: extended_enat,Y: extended_enat,Nb: nat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(Xb,Y)),extended_enat2(Nb))
    <=> ? [Y7: nat,X8: nat] :
          ( ( Xb = extended_enat2(X8) )
          & ( Y = extended_enat2(Y7) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X8),Y7)),Nb) ) ) ).

% iadd_le_enat_iff
tff(fact_10116_elimnum,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)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Nb))) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimnum
tff(fact_10117_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT,L: nat] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extended_enat2(L)) = vEBT_Node(Info,Dega,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Dega),TreeLista)),vEBT_VEBT_elim_dead(Summarya,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Dega),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ).

% VEBT_internal.elim_dead.simps(3)
tff(fact_10118_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( ! [A4: $o,B4: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B4)) )
           => ( Y != vEBT_Leaf((A4),(B4)) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
             => ( ( Xaa = extend5688581933313929465d_enat )
               => ( Y != vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ! [L2: nat] :
                    ( ( Xaa = extended_enat2(L2) )
                   => ( Y != vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_10119_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Dega: nat,TreeLista: list_VEBT_VEBT,Summarya: vEBT_VEBT] : vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Info,Dega,map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Dega),TreeLista),vEBT_VEBT_elim_dead(Summarya,extend5688581933313929465d_enat)) ).

% VEBT_internal.elim_dead.simps(2)
tff(fact_10120_elimcomplete,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)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Dega,TreeLista,Summarya),extend5688581933313929465d_enat) = vEBT_Node(Info,Dega,TreeLista,Summarya) ) ) ).

% elimcomplete
tff(fact_10121_not__infinity__eq,axiom,
    ! [Xb: extended_enat] :
      ( ( Xb != extend5688581933313929465d_enat )
    <=> ? [I2: nat] : Xb = extended_enat2(I2) ) ).

% not_infinity_eq
tff(fact_10122_not__enat__eq,axiom,
    ! [Xb: extended_enat] :
      ( ! [Y5: nat] : Xb != extended_enat2(Y5)
    <=> ( Xb = extend5688581933313929465d_enat ) ) ).

% not_enat_eq
tff(fact_10123_enat__ord__simps_I4_J,axiom,
    ! [Q2: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Q2),extend5688581933313929465d_enat)
    <=> ( Q2 != extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(4)
tff(fact_10124_enat__ord__simps_I6_J,axiom,
    ! [Q2: extended_enat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),Q2) ).

% enat_ord_simps(6)
tff(fact_10125_plus__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : plus_p3455044024723400733d_enat(extend5688581933313929465d_enat,Q2) = extend5688581933313929465d_enat ).

% plus_enat_simps(2)
tff(fact_10126_plus__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : plus_p3455044024723400733d_enat(Q2,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% plus_enat_simps(3)
tff(fact_10127_enat__ord__code_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Q2),extend5688581933313929465d_enat) ).

% enat_ord_code(3)
tff(fact_10128_enat__ord__simps_I5_J,axiom,
    ! [Q2: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),Q2)
    <=> ( Q2 = extend5688581933313929465d_enat ) ) ).

% enat_ord_simps(5)
tff(fact_10129_idiff__infinity,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(extend5688581933313929465d_enat,Nb) = extend5688581933313929465d_enat ).

% idiff_infinity
tff(fact_10130_times__enat__simps_I2_J,axiom,
    times_7803423173614009249d_enat(extend5688581933313929465d_enat,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% times_enat_simps(2)
tff(fact_10131_max__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Q2),extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% max_enat_simps(4)
tff(fact_10132_max__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,extend5688581933313929465d_enat),Q2) = extend5688581933313929465d_enat ).

% max_enat_simps(5)
tff(fact_10133_min__enat__simps_I4_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,Q2),extend5688581933313929465d_enat) = Q2 ).

% min_enat_simps(4)
tff(fact_10134_min__enat__simps_I5_J,axiom,
    ! [Q2: extended_enat] : aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_mi8085742599997312461d_enat,extend5688581933313929465d_enat),Q2) = Q2 ).

% min_enat_simps(5)
tff(fact_10135_idiff__self,axiom,
    ! [Nb: extended_enat] :
      ( ( Nb != extend5688581933313929465d_enat )
     => ( minus_3235023915231533773d_enat(Nb,Nb) = zero_z5237406670263579293d_enat ) ) ).

% idiff_self
tff(fact_10136_add__diff__cancel__enat,axiom,
    ! [Xb: extended_enat,Y: extended_enat] :
      ( ( Xb != extend5688581933313929465d_enat )
     => ( minus_3235023915231533773d_enat(plus_p3455044024723400733d_enat(Xb,Y),Xb) = Y ) ) ).

% add_diff_cancel_enat
tff(fact_10137_idiff__infinity__right,axiom,
    ! [A: nat] : minus_3235023915231533773d_enat(extended_enat2(A),extend5688581933313929465d_enat) = zero_z5237406670263579293d_enat ).

% idiff_infinity_right
tff(fact_10138_times__enat__simps_I3_J,axiom,
    ! [Nb: nat] :
      times_7803423173614009249d_enat(extend5688581933313929465d_enat,extended_enat2(Nb)) = $ite(Nb = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(3)
tff(fact_10139_times__enat__simps_I4_J,axiom,
    ! [Mb: nat] :
      times_7803423173614009249d_enat(extended_enat2(Mb),extend5688581933313929465d_enat) = $ite(Mb = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat) ).

% times_enat_simps(4)
tff(fact_10140_Sup__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      comple4398354569131411667d_enat(A3) = $ite(
        A3 = bot_bo7653980558646680370d_enat,
        zero_z5237406670263579293d_enat,
        $ite(finite4001608067531595151d_enat(A3),lattic921264341876707157d_enat(A3),extend5688581933313929465d_enat) ) ).

% Sup_enat_def
tff(fact_10141_Inf__enat__def,axiom,
    ! [A3: set_Extended_enat] :
      comple2295165028678016749d_enat(A3) = $ite(A3 = bot_bo7653980558646680370d_enat,extend5688581933313929465d_enat,ord_Le1955565732374568822d_enat(aTP_Lamm_a(A3))) ).

% Inf_enat_def
tff(fact_10142_imult__is__infinity,axiom,
    ! [A: extended_enat,B2: extended_enat] :
      ( ( times_7803423173614009249d_enat(A,B2) = extend5688581933313929465d_enat )
    <=> ( ( ( A = extend5688581933313929465d_enat )
          & ( B2 != zero_z5237406670263579293d_enat ) )
        | ( ( B2 = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
tff(fact_10143_infinity__ne__i0,axiom,
    extend5688581933313929465d_enat != zero_z5237406670263579293d_enat ).

% infinity_ne_i0
tff(fact_10144_enat__add__left__cancel__less,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(plus_p3455044024723400733d_enat(A,B2)),plus_p3455044024723400733d_enat(A,C))
    <=> ( ( A != extend5688581933313929465d_enat )
        & aa_Extended_enat_o(ord_le72135733267957522d_enat(B2),C) ) ) ).

% enat_add_left_cancel_less
tff(fact_10145_infinity__ne__i1,axiom,
    extend5688581933313929465d_enat != one_on7984719198319812577d_enat ).

% infinity_ne_i1
tff(fact_10146_plus__eq__infty__iff__enat,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(Mb,Nb) = extend5688581933313929465d_enat )
    <=> ( ( Mb = extend5688581933313929465d_enat )
        | ( Nb = extend5688581933313929465d_enat ) ) ) ).

% plus_eq_infty_iff_enat
tff(fact_10147_enat__add__left__cancel,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( ( plus_p3455044024723400733d_enat(A,B2) = plus_p3455044024723400733d_enat(A,C) )
    <=> ( ( A = extend5688581933313929465d_enat )
        | ( B2 = C ) ) ) ).

% enat_add_left_cancel
tff(fact_10148_numeral__ne__infinity,axiom,
    ! [K: num] : numera1916890842035813515d_enat(K) != extend5688581933313929465d_enat ).

% numeral_ne_infinity
tff(fact_10149_top__enat__def,axiom,
    top_to3028658606643905974d_enat = extend5688581933313929465d_enat ).

% top_enat_def
tff(fact_10150_enat__ord__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Q2),extend5688581933313929465d_enat) ).

% enat_ord_simps(3)
tff(fact_10151_enat__add__left__cancel__le,axiom,
    ! [A: extended_enat,B2: extended_enat,C: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(plus_p3455044024723400733d_enat(A,B2)),plus_p3455044024723400733d_enat(A,C))
    <=> ( ( A = extend5688581933313929465d_enat )
        | aa_Extended_enat_o(ord_le2932123472753598470d_enat(B2),C) ) ) ).

% enat_add_left_cancel_le
tff(fact_10152_enat__ord__code_I5_J,axiom,
    ! [Nb: nat] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),extended_enat2(Nb)) ).

% enat_ord_code(5)
tff(fact_10153_infinity__ileE,axiom,
    ! [Mb: nat] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(extend5688581933313929465d_enat),extended_enat2(Mb)) ).

% infinity_ileE
tff(fact_10154_enat__ex__split,axiom,
    ! [P: fun_Extended_enat_o] :
      ( ? [X_13: extended_enat] : aa_Extended_enat_o(P,X_13)
    <=> ( aa_Extended_enat_o(P,extend5688581933313929465d_enat)
        | ? [X: nat] : aa_Extended_enat_o(P,extended_enat2(X)) ) ) ).

% enat_ex_split
tff(fact_10155_enat3__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
         => ! [Natb: nat] : Yb != extended_enat2(Natb) ) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( ? [Nata: nat] : Ya = extended_enat2(Nata)
           => ( Yb != extend5688581933313929465d_enat ) ) )
       => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
           => ( ( Ya = extend5688581933313929465d_enat )
             => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
         => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
             => ( ( Ya = extend5688581933313929465d_enat )
               => ( Yb != extend5688581933313929465d_enat ) ) )
           => ( ( ( Y = extend5688581933313929465d_enat )
               => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                 => ! [Nata: nat] : Yb != extended_enat2(Nata) ) )
             => ( ( ( Y = extend5688581933313929465d_enat )
                 => ( ? [Nat3: nat] : Ya = extended_enat2(Nat3)
                   => ( Yb != extend5688581933313929465d_enat ) ) )
               => ( ( ( Y = extend5688581933313929465d_enat )
                   => ( ( Ya = extend5688581933313929465d_enat )
                     => ! [Nat3: nat] : Yb != extended_enat2(Nat3) ) )
                 => ~ ( ( Y = extend5688581933313929465d_enat )
                     => ( ( Ya = extend5688581933313929465d_enat )
                       => ( Yb != extend5688581933313929465d_enat ) ) ) ) ) ) ) ) ) ) ).

% enat3_cases
tff(fact_10156_enat2__cases,axiom,
    ! [Y: extended_enat,Ya: extended_enat] :
      ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
       => ! [Nata: nat] : Ya != extended_enat2(Nata) )
     => ( ( ? [Nat3: nat] : Y = extended_enat2(Nat3)
         => ( Ya != extend5688581933313929465d_enat ) )
       => ( ( ( Y = extend5688581933313929465d_enat )
           => ! [Nat3: nat] : Ya != extended_enat2(Nat3) )
         => ~ ( ( Y = extend5688581933313929465d_enat )
             => ( Ya != extend5688581933313929465d_enat ) ) ) ) ) ).

% enat2_cases
tff(fact_10157_enat_Oexhaust,axiom,
    ! [Y: extended_enat] :
      ( ! [Nat3: nat] : Y != extended_enat2(Nat3)
     => ( Y = extend5688581933313929465d_enat ) ) ).

% enat.exhaust
tff(fact_10158_enat_Odistinct_I1_J,axiom,
    ! [Nat: nat] : extended_enat2(Nat) != extend5688581933313929465d_enat ).

% enat.distinct(1)
tff(fact_10159_enat_Odistinct_I2_J,axiom,
    ! [Nat: nat] : extend5688581933313929465d_enat != extended_enat2(Nat) ).

% enat.distinct(2)
tff(fact_10160_infinity__ilessE,axiom,
    ! [Mb: nat] : ~ aa_Extended_enat_o(ord_le72135733267957522d_enat(extend5688581933313929465d_enat),extended_enat2(Mb)) ).

% infinity_ilessE
tff(fact_10161_less__infinityE,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),extend5688581933313929465d_enat)
     => ~ ! [K2: nat] : Nb != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_10162_enat__ord__code_I4_J,axiom,
    ! [Mb: nat] : aa_Extended_enat_o(ord_le72135733267957522d_enat(extended_enat2(Mb)),extend5688581933313929465d_enat) ).

% enat_ord_code(4)
tff(fact_10163_imult__infinity,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
     => ( times_7803423173614009249d_enat(extend5688581933313929465d_enat,Nb) = extend5688581933313929465d_enat ) ) ).

% imult_infinity
tff(fact_10164_imult__infinity__right,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),Nb)
     => ( times_7803423173614009249d_enat(Nb,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
tff(fact_10165_plus__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : plus_p3455044024723400733d_enat(Mb,Nb) = extend3600170679010898289d_enat(aTP_Lamm_mc(Nb),extend5688581933313929465d_enat,Mb) ).

% plus_enat_def
tff(fact_10166_diff__enat__def,axiom,
    ! [A: extended_enat,B2: extended_enat] : minus_3235023915231533773d_enat(A,B2) = extend3600170679010898289d_enat(aTP_Lamm_me(B2),extend5688581933313929465d_enat,A) ).

% diff_enat_def
tff(fact_10167_times__enat__def,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      times_7803423173614009249d_enat(Mb,Nb) = extend3600170679010898289d_enat(aTP_Lamm_mg(Nb),
        $ite(Nb = zero_z5237406670263579293d_enat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),
        Mb) ).

% times_enat_def
tff(fact_10168_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: extended_enat,Y: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xb,Xaa) = Y )
     => ( accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(Xb,Xaa))
       => ( ! [A4: $o,B4: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B4)) )
             => ( ( Y = vEBT_Leaf((A4),(B4)) )
               => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Leaf((A4),(B4)),Xaa)) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
               => ( ( Xaa = extend5688581933313929465d_enat )
                 => ( ( Y = vEBT_Node(Info2,Deg,map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Deg),TreeList),vEBT_VEBT_elim_dead(Summary,extend5688581933313929465d_enat)) )
                   => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg,TreeList,Summary),extend5688581933313929465d_enat)) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,Deg,TreeList,Summary) )
                 => ! [L2: nat] :
                      ( ( Xaa = extended_enat2(L2) )
                     => ( ( Y = vEBT_Node(Info2,Deg,take_VEBT_VEBT(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))),map_VE8901447254227204932T_VEBT(aTP_Lamm_ma(Deg),TreeList)),vEBT_VEBT_elim_dead(Summary,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,L2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))))) )
                       => ~ accp_P6183159247885693666d_enat(vEBT_V312737461966249ad_rel,produc581526299967858633d_enat(vEBT_Node(Info2,Deg,TreeList,Summary),extended_enat2(L2))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_10169_the__enat_Osimps,axiom,
    ! [Nb: nat] : aa_Extended_enat_nat(extended_the_enat,extended_enat2(Nb)) = Nb ).

% the_enat.simps
tff(fact_10170_eSuc__Max,axiom,
    ! [A3: set_Extended_enat] :
      ( finite4001608067531595151d_enat(A3)
     => ( ( A3 != bot_bo7653980558646680370d_enat )
       => ( aa_Ext7969882492699152700d_enat(extended_eSuc,lattic921264341876707157d_enat(A3)) = lattic921264341876707157d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ) ).

% eSuc_Max
tff(fact_10171_eSuc__inject,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,Mb) = aa_Ext7969882492699152700d_enat(extended_eSuc,Nb) )
    <=> ( Mb = Nb ) ) ).

% eSuc_inject
tff(fact_10172_eSuc__infinity,axiom,
    aa_Ext7969882492699152700d_enat(extended_eSuc,extend5688581933313929465d_enat) = extend5688581933313929465d_enat ).

% eSuc_infinity
tff(fact_10173_eSuc__mono,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)),aa_Ext7969882492699152700d_enat(extended_eSuc,Mb))
    <=> aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),Mb) ) ).

% eSuc_mono
tff(fact_10174_eSuc__ile__mono,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] :
      ( aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)),aa_Ext7969882492699152700d_enat(extended_eSuc,Mb))
    <=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),Mb) ) ).

% eSuc_ile_mono
tff(fact_10175_eSuc__minus__eSuc,axiom,
    ! [Nb: extended_enat,Mb: extended_enat] : minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Nb),aa_Ext7969882492699152700d_enat(extended_eSuc,Mb)) = minus_3235023915231533773d_enat(Nb,Mb) ).

% eSuc_minus_eSuc
tff(fact_10176_iless__eSuc0,axiom,
    ! [Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Nb),aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat))
    <=> ( Nb = zero_z5237406670263579293d_enat ) ) ).

% iless_eSuc0
tff(fact_10177_eSuc__minus__1,axiom,
    ! [Nb: extended_enat] : minus_3235023915231533773d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Nb),one_on7984719198319812577d_enat) = Nb ).

% eSuc_minus_1
tff(fact_10178_eSuc__numeral,axiom,
    ! [K: num] : aa_Ext7969882492699152700d_enat(extended_eSuc,numera1916890842035813515d_enat(K)) = numera1916890842035813515d_enat(aa_num_num(plus_plus_num(K),one)) ).

% eSuc_numeral
tff(fact_10179_iless__Suc__eq,axiom,
    ! [Mb: nat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(extended_enat2(Mb)),aa_Ext7969882492699152700d_enat(extended_eSuc,Nb))
    <=> aa_Extended_enat_o(ord_le2932123472753598470d_enat(extended_enat2(Mb)),Nb) ) ).

% iless_Suc_eq
tff(fact_10180_one__eSuc,axiom,
    one_on7984719198319812577d_enat = aa_Ext7969882492699152700d_enat(extended_eSuc,zero_z5237406670263579293d_enat) ).

% one_eSuc
tff(fact_10181_i0__iless__eSuc,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le72135733267957522d_enat(zero_z5237406670263579293d_enat),aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)) ).

% i0_iless_eSuc
tff(fact_10182_zero__ne__eSuc,axiom,
    ! [Nb: extended_enat] : zero_z5237406670263579293d_enat != aa_Ext7969882492699152700d_enat(extended_eSuc,Nb) ).

% zero_ne_eSuc
tff(fact_10183_eSuc__plus__1,axiom,
    ! [Nb: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,Nb) = plus_p3455044024723400733d_enat(Nb,one_on7984719198319812577d_enat) ).

% eSuc_plus_1
tff(fact_10184_plus__1__eSuc_I1_J,axiom,
    ! [Q2: extended_enat] : plus_p3455044024723400733d_enat(one_on7984719198319812577d_enat,Q2) = aa_Ext7969882492699152700d_enat(extended_eSuc,Q2) ).

% plus_1_eSuc(1)
tff(fact_10185_plus__1__eSuc_I2_J,axiom,
    ! [Q2: extended_enat] : plus_p3455044024723400733d_enat(Q2,one_on7984719198319812577d_enat) = aa_Ext7969882492699152700d_enat(extended_eSuc,Q2) ).

% plus_1_eSuc(2)
tff(fact_10186_mono__eSuc,axiom,
    order_4130057895858720880d_enat(extended_eSuc) ).

% mono_eSuc
tff(fact_10187_iadd__Suc__right,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : plus_p3455044024723400733d_enat(Mb,aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)) = aa_Ext7969882492699152700d_enat(extended_eSuc,plus_p3455044024723400733d_enat(Mb,Nb)) ).

% iadd_Suc_right
tff(fact_10188_iadd__Suc,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : plus_p3455044024723400733d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Mb),Nb) = aa_Ext7969882492699152700d_enat(extended_eSuc,plus_p3455044024723400733d_enat(Mb,Nb)) ).

% iadd_Suc
tff(fact_10189_eSuc__max,axiom,
    ! [Xb: extended_enat,Y: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,Xb),Y)) = aa_Ext7969882492699152700d_enat(aa_Ext3475206891998675565d_enat(ord_ma741700101516333627d_enat,aa_Ext7969882492699152700d_enat(extended_eSuc,Xb)),aa_Ext7969882492699152700d_enat(extended_eSuc,Y)) ).

% eSuc_max
tff(fact_10190_mult__eSuc__right,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : times_7803423173614009249d_enat(Mb,aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)) = plus_p3455044024723400733d_enat(Mb,times_7803423173614009249d_enat(Mb,Nb)) ).

% mult_eSuc_right
tff(fact_10191_mult__eSuc,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] : times_7803423173614009249d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Mb),Nb) = plus_p3455044024723400733d_enat(Nb,times_7803423173614009249d_enat(Mb,Nb)) ).

% mult_eSuc
tff(fact_10192_ileI1,axiom,
    ! [Mb: extended_enat,Nb: extended_enat] :
      ( aa_Extended_enat_o(ord_le72135733267957522d_enat(Mb),Nb)
     => aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Mb)),Nb) ) ).

% ileI1
tff(fact_10193_ile__eSuc,axiom,
    ! [Nb: extended_enat] : aa_Extended_enat_o(ord_le2932123472753598470d_enat(Nb),aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)) ).

% ile_eSuc
tff(fact_10194_not__eSuc__ilei0,axiom,
    ! [Nb: extended_enat] : ~ aa_Extended_enat_o(ord_le2932123472753598470d_enat(aa_Ext7969882492699152700d_enat(extended_eSuc,Nb)),zero_z5237406670263579293d_enat) ).

% not_eSuc_ilei0
tff(fact_10195_eSuc__enat,axiom,
    ! [Nb: nat] : aa_Ext7969882492699152700d_enat(extended_eSuc,extended_enat2(Nb)) = extended_enat2(aa_nat_nat(suc,Nb)) ).

% eSuc_enat
tff(fact_10196_eSuc__enat__iff,axiom,
    ! [Xb: extended_enat,Y: nat] :
      ( ( aa_Ext7969882492699152700d_enat(extended_eSuc,Xb) = extended_enat2(Y) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( Xb = extended_enat2(N4) ) ) ) ).

% eSuc_enat_iff
tff(fact_10197_enat__eSuc__iff,axiom,
    ! [Y: nat,Xb: extended_enat] :
      ( ( extended_enat2(Y) = aa_Ext7969882492699152700d_enat(extended_eSuc,Xb) )
    <=> ? [N4: nat] :
          ( ( Y = aa_nat_nat(suc,N4) )
          & ( extended_enat2(N4) = Xb ) ) ) ).

% enat_eSuc_iff
tff(fact_10198_eSuc__Sup,axiom,
    ! [A3: set_Extended_enat] :
      ( ( A3 != bot_bo7653980558646680370d_enat )
     => ( aa_Ext7969882492699152700d_enat(extended_eSuc,comple4398354569131411667d_enat(A3)) = comple4398354569131411667d_enat(image_80655429650038917d_enat(extended_eSuc,A3)) ) ) ).

% eSuc_Sup
tff(fact_10199_eSuc__def,axiom,
    ! [I: extended_enat] : aa_Ext7969882492699152700d_enat(extended_eSuc,I) = extend3600170679010898289d_enat(aTP_Lamm_mh,extend5688581933313929465d_enat,I) ).

% eSuc_def
tff(fact_10200_less__than__iff,axiom,
    ! [Xb: nat,Y: nat] :
      ( member8440522571783428010at_nat(aa_nat7603221163087113075at_nat(product_Pair_nat_nat(Xb),Y),less_than)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Xb),Y) ) ).

% less_than_iff
tff(fact_10201_Quotient__real,axiom,
    quotie6870213296501934825t_real(realrel,real2,rep_real,cr_real) ).

% Quotient_real
tff(fact_10202_lcm__code__integer,axiom,
    ! [A: code_integer,B2: code_integer] : gcd_lcm_Code_integer(A,B2) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B2)),gcd_gcd_Code_integer(A,B2)) ).

% lcm_code_integer
tff(fact_10203_lcm__1__iff__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Mb),Nb) = aa_nat_nat(suc,zero_zero_nat) )
    <=> ( ( Mb = aa_nat_nat(suc,zero_zero_nat) )
        & ( Nb = aa_nat_nat(suc,zero_zero_nat) ) ) ) ).

% lcm_1_iff_nat
tff(fact_10204_prod__gcd__lcm__int,axiom,
    ! [Mb: int,Nb: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Mb)),aa_int_int(abs_abs_int,Nb)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(gcd_gcd_int(Mb),Nb)),aa_int_int(gcd_lcm_int(Mb),Nb)) ).

% prod_gcd_lcm_int
tff(fact_10205_prod__gcd__lcm__nat,axiom,
    ! [Mb: nat,Nb: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Mb),Nb) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Mb),Nb)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Mb),Nb)) ).

% prod_gcd_lcm_nat
tff(fact_10206_lcm__nat__def,axiom,
    ! [Xb: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_lcm_nat,Xb),Y) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Xb),Y)),aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,Xb),Y)) ).

% lcm_nat_def
tff(fact_10207_lcm__pos__nat,axiom,
    ! [Mb: nat,Nb: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),Mb)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,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(gcd_lcm_nat,Mb),Nb)) ) ) ).

% lcm_pos_nat
tff(fact_10208_lcm__pos__int,axiom,
    ! [Mb: int,Nb: int] :
      ( ( Mb != zero_zero_int )
     => ( ( Nb != zero_zero_int )
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(gcd_lcm_int(Mb),Nb)) ) ) ).

% lcm_pos_int
tff(fact_10209_lcm__altdef__int,axiom,
    ! [A: int,B2: int] : aa_int_int(gcd_lcm_int(A),B2) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B2))),aa_int_int(gcd_gcd_int(A),B2)) ).

% lcm_altdef_int
tff(fact_10210_ATP_Olambda__1,axiom,
    ! [Uu2: fun_nat_rat] :
      aa_fun765758790203053448at_rat(aTP_Lamm_lf,Uu2) = $ite(vanishes(Uu2),aTP_Lamm_ld,aTP_Lamm_kz(Uu2)) ).

% ATP.lambda_1
tff(fact_10211_ATP_Olambda__2,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_dt,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uu2)),one_one_real)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_2
tff(fact_10212_ATP_Olambda__3,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_eb,Uu2) = 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,Uu2)) ).

% ATP.lambda_3
tff(fact_10213_ATP_Olambda__4,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_fr,Uu2)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,zero_zero_real),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_eq_real,Uu2),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))
        & ( aa_real_real(cos_real,Uu2) = zero_zero_real ) ) ) ).

% ATP.lambda_4
tff(fact_10214_ATP_Olambda__5,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_eq,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uu2)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uu2)) ).

% ATP.lambda_5
tff(fact_10215_ATP_Olambda__6,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_iy,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(cos_real,Uu2)),aa_real_real(sin_real,Uu2)) ).

% ATP.lambda_6
tff(fact_10216_ATP_Olambda__7,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_jy,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(ln_ln_real,Uu2)),Uu2) ).

% ATP.lambda_7
tff(fact_10217_ATP_Olambda__8,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_ja,Uu2) = aa_real_real(aa_nat_fun_real_real(root,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_8
tff(fact_10218_ATP_Olambda__9,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_hz,Uu2) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_9
tff(fact_10219_ATP_Olambda__10,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_li,Uu2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),Uu2) ).

% ATP.lambda_10
tff(fact_10220_ATP_Olambda__11,axiom,
    ! [Uu2: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ll,Uu2)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uu2))) ) ).

% ATP.lambda_11
tff(fact_10221_ATP_Olambda__12,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_jd,Uu2) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,Uu2)) ).

% ATP.lambda_12
tff(fact_10222_ATP_Olambda__13,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_iw,Uu2) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_dq,Uu2)) ).

% ATP.lambda_13
tff(fact_10223_ATP_Olambda__14,axiom,
    ! [Uu2: real] : aa_rea5117059003799598681omplex(aTP_Lamm_kq,Uu2) = princi3496590319149328850omplex(collec8663557070575231912omplex(produc6771430404735790350plex_o(aTP_Lamm_kp(Uu2)))) ).

% ATP.lambda_14
tff(fact_10224_ATP_Olambda__15,axiom,
    ! [Uu2: real] : aa_rea5428163979925208277l_real(aTP_Lamm_ko,Uu2) = princi6114159922880469582l_real(collec3799799289383736868l_real(produc5414030515140494994real_o(aTP_Lamm_kn(Uu2)))) ).

% ATP.lambda_15
tff(fact_10225_ATP_Olambda__16,axiom,
    ! [Uu2: nat] : aa_nat_real(aTP_Lamm_jf,Uu2) = aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uu2))) ).

% ATP.lambda_16
tff(fact_10226_ATP_Olambda__17,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_hg,Uu2) = some_num(aa_num_num(bit1,Uu2)) ).

% ATP.lambda_17
tff(fact_10227_ATP_Olambda__18,axiom,
    ! [Uu2: num] : aa_num_option_num(aTP_Lamm_hf,Uu2) = some_num(aa_num_num(bit0,Uu2)) ).

% ATP.lambda_18
tff(fact_10228_ATP_Olambda__19,axiom,
    ! [Uu2: nat] : aa_nat8803788312118491388at_nat(aTP_Lamm_fn,Uu2) = product_Pair_nat_nat(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_19
tff(fact_10229_ATP_Olambda__20,axiom,
    ! [Uu2: nat] : aa_nat_Extended_enat(aTP_Lamm_mh,Uu2) = extended_enat2(aa_nat_nat(suc,Uu2)) ).

% ATP.lambda_20
tff(fact_10230_ATP_Olambda__21,axiom,
    ! [Uu2: fun_nat_rat] :
      ( aa_fun_nat_rat_o(aTP_Lamm_lg,Uu2)
    <=> ? [R5: rat] :
          ( aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,zero_zero_rat),R5)
          & ? [K3: nat] :
            ! [N4: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,K3),N4)
             => aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,R5),aa_nat_rat(Uu2,N4)) ) ) ) ).

% ATP.lambda_21
tff(fact_10231_ATP_Olambda__22,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_ib,Uu2)
    <=> ? [I2: int,N4: nat] :
          ( ( Uu2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I2)),aa_nat_real(semiri5074537144036343181t_real,N4)) )
          & ( N4 != zero_zero_nat ) ) ) ).

% ATP.lambda_22
tff(fact_10232_ATP_Olambda__23,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_ia,Uu2)
    <=> ? [I2: int,J3: int] :
          ( ( Uu2 = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_int_real(ring_1_of_int_real,I2)),aa_int_real(ring_1_of_int_real,J3)) )
          & ( J3 != zero_zero_int ) ) ) ).

% ATP.lambda_23
tff(fact_10233_ATP_Olambda__24,axiom,
    ! [Uu2: nat] : aa_nat_option_num(aTP_Lamm_he,Uu2) = some_num(one) ).

% ATP.lambda_24
tff(fact_10234_ATP_Olambda__25,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_hm(Uu2),Uua) = case_num_option_num(some_num(one),aTP_Lamm_hk(Uua),aTP_Lamm_hl(Uua),Uu2) ).

% ATP.lambda_25
tff(fact_10235_ATP_Olambda__26,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] :
      aa_nat_real(aTP_Lamm_ec(Uu2),Uua) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua),zero_zero_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_26
tff(fact_10236_ATP_Olambda__27,axiom,
    ! [Uu2: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gf,Uu2),Uua) = $ite(Uu2 = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,Uu2)),Uua),aa_int_int(abs_abs_int,Uu2))) ).

% ATP.lambda_27
tff(fact_10237_ATP_Olambda__28,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      aa_nat_Extended_enat(aTP_Lamm_mg(Uu2),Uua) = extend3600170679010898289d_enat(aTP_Lamm_mf(Uua),
        $ite(Uua = zero_zero_nat,zero_z5237406670263579293d_enat,extend5688581933313929465d_enat),
        Uu2) ).

% ATP.lambda_28
tff(fact_10238_ATP_Olambda__29,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_me(Uu2),Uua) = extend3600170679010898289d_enat(aTP_Lamm_md(Uua),zero_z5237406670263579293d_enat,Uu2) ).

% ATP.lambda_29
tff(fact_10239_ATP_Olambda__30,axiom,
    ! [Uu2: extended_enat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_mc(Uu2),Uua) = extend3600170679010898289d_enat(aTP_Lamm_mb(Uua),extend5688581933313929465d_enat,Uu2) ).

% ATP.lambda_30
tff(fact_10240_ATP_Olambda__31,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_lr(Uu2),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(aTP_Lamm_eh,Uua),$false),Uu2) ) ).

% ATP.lambda_31
tff(fact_10241_ATP_Olambda__32,axiom,
    ! [Uu2: extended_enat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ls(Uu2),Uua)
    <=> aa_Extended_enat_o(extended_case_enat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),$true),Uu2) ) ).

% ATP.lambda_32
tff(fact_10242_ATP_Olambda__33,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aa_nat3713591252005393295on_num(aTP_Lamm_hn,Uu2),Uua) = case_nat_option_num(none_num,aTP_Lamm_hm(Uua),Uu2) ).

% ATP.lambda_33
tff(fact_10243_ATP_Olambda__34,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_hk(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_hf,bit_take_bit_num(Uu2,Uua)) ).

% ATP.lambda_34
tff(fact_10244_ATP_Olambda__35,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_hh(Uu2),Uua) = case_o6005452278849405969um_num(none_num,aTP_Lamm_hf,bit_take_bit_num(Uua,Uu2)) ).

% ATP.lambda_35
tff(fact_10245_ATP_Olambda__36,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_lj,Uu2),Uua) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_snd_int_int(Uua))),aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uua)),product_snd_int_int(Uu2))),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_36
tff(fact_10246_ATP_Olambda__37,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dp,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),one_one_nat))))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),one_one_real)),aa_nat_nat(suc,Uua))) ).

% ATP.lambda_37
tff(fact_10247_ATP_Olambda__38,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ea,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_38
tff(fact_10248_ATP_Olambda__39,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ee,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_39
tff(fact_10249_ATP_Olambda__40,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fs,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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) = Uu2 ) ) ) ).

% ATP.lambda_40
tff(fact_10250_ATP_Olambda__41,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fo,Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(uminus_uminus_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,pi),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) = Uu2 ) ) ) ).

% ATP.lambda_41
tff(fact_10251_ATP_Olambda__42,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_gp,Uu2),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),code_int_of_integer(Uu2)),
        $ite(Uua = zero_z3403309356797280102nteger,l2,aa_int_int(aa_int_fun_int_int(plus_plus_int,l2),one_one_int)) ) ).

% ATP.lambda_42
tff(fact_10252_ATP_Olambda__43,axiom,
    ! [Uu2: complex,Uua: real] :
      ( aa_real_o(aTP_Lamm_fk(Uu2),Uua)
    <=> ( ( aa_complex_complex(sgn_sgn_complex,Uu2) = 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_43
tff(fact_10253_ATP_Olambda__44,axiom,
    ! [Uu2: product_prod_int_int,Uua: product_prod_int_int] : aa_Pro6970504127699319548nt_int(aa_Pro4124234152028750080nt_int(aTP_Lamm_lk,Uu2),Uua) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,product_fst_int_int(Uu2)),product_fst_int_int(Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,product_snd_int_int(Uu2)),product_snd_int_int(Uua))) ).

% ATP.lambda_44
tff(fact_10254_ATP_Olambda__45,axiom,
    ! [Uu2: real,Uua: int] :
      ( aa_int_o(aTP_Lamm_gd(Uu2),Uua)
    <=> ( aa_real_o(aa_real_fun_real_o(ord_less_eq_real,aa_int_real(ring_1_of_int_real,Uua)),Uu2)
        & aa_real_o(aa_real_fun_real_o(ord_less_real,Uu2),aa_int_real(ring_1_of_int_real,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_45
tff(fact_10255_ATP_Olambda__46,axiom,
    ! [Uu2: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_ge(Uu2),Uua)
    <=> ( aa_rat_o(aa_rat_fun_rat_o(ord_less_eq_rat,aa_int_rat(ring_1_of_int_rat,Uua)),Uu2)
        & aa_rat_o(aa_rat_fun_rat_o(ord_less_rat,Uu2),aa_int_rat(ring_1_of_int_rat,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_46
tff(fact_10256_ATP_Olambda__47,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dq,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))) ).

% ATP.lambda_47
tff(fact_10257_ATP_Olambda__48,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ix,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_48
tff(fact_10258_ATP_Olambda__49,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_iz(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(uminus_uminus_real,one_one_real)),Uua)),aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_49
tff(fact_10259_ATP_Olambda__50,axiom,
    ! [Uu2: rat,Uua: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_ht(Uu2),Uua)
    <=> ( ( Uu2 = aa_int_rat(fract(product_fst_int_int(Uua)),product_snd_int_int(Uua)) )
        & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),product_snd_int_int(Uua))
        & aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,product_fst_int_int(Uua)),product_snd_int_int(Uua)) ) ) ).

% ATP.lambda_50
tff(fact_10260_ATP_Olambda__51,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fh,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))) ).

% ATP.lambda_51
tff(fact_10261_ATP_Olambda__52,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ew,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uua)) ).

% ATP.lambda_52
tff(fact_10262_ATP_Olambda__53,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_jz,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2)),aa_real_real(exp_real,Uua)) ).

% ATP.lambda_53
tff(fact_10263_ATP_Olambda__54,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_lz(Uu2),Uua) = aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_54
tff(fact_10264_ATP_Olambda__55,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hx,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_55
tff(fact_10265_ATP_Olambda__56,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fa,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uu2) ).

% ATP.lambda_56
tff(fact_10266_ATP_Olambda__57,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ez,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)),Uua) ).

% ATP.lambda_57
tff(fact_10267_ATP_Olambda__58,axiom,
    ! [Uu2: nat,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_er(Uu2),Uua)
    <=> ( aa_nat_complex(power_power_complex(Uua),Uu2) = one_one_complex ) ) ).

% ATP.lambda_58
tff(fact_10268_ATP_Olambda__59,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jk,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)))),Uua) ).

% ATP.lambda_59
tff(fact_10269_ATP_Olambda__60,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_kc,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),Uua))),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),Uua)) ).

% ATP.lambda_60
tff(fact_10270_ATP_Olambda__61,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ka,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uu2),Uua))),Uua) ).

% ATP.lambda_61
tff(fact_10271_ATP_Olambda__62,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bt,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,one_one_real),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),one_one_nat))) ).

% ATP.lambda_62
tff(fact_10272_ATP_Olambda__63,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_fq,Uu2),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) = Uu2 ) ) ) ).

% ATP.lambda_63
tff(fact_10273_ATP_Olambda__64,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cx(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_real(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_64
tff(fact_10274_ATP_Olambda__65,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_cy(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_rat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_65
tff(fact_10275_ATP_Olambda__66,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_cz(Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_nat(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_66
tff(fact_10276_ATP_Olambda__67,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_da(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))),aa_nat_int(Uu2,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)))) ).

% ATP.lambda_67
tff(fact_10277_ATP_Olambda__68,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat_complex(aTP_Lamm_dr(Uu2),Uua) = aa_complex_complex(times_times_complex(aa_nat_complex(Uu2,Uua)),aa_nat_complex(power_power_complex(zero_zero_complex),Uua)) ).

% ATP.lambda_68
tff(fact_10278_ATP_Olambda__69,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ds(Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,zero_zero_real),Uua)) ).

% ATP.lambda_69
tff(fact_10279_ATP_Olambda__70,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jq(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_iz(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat))) ).

% ATP.lambda_70
tff(fact_10280_ATP_Olambda__71,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jp(Uu2),Uua) = groups6591440286371151544t_real(aTP_Lamm_iz(Uu2),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ).

% ATP.lambda_71
tff(fact_10281_ATP_Olambda__72,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ha,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(nat_triangle,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua))),Uu2) ).

% ATP.lambda_72
tff(fact_10282_ATP_Olambda__73,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aTP_Lamm_hd(Uu2),Uua) = aa_int_int(bit_se2923211474154528505it_int(aa_num_nat(numeral_numeral_nat,Uu2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),aa_num_nat(numeral_numeral_nat,Uu2))),aa_num_int(numeral_numeral_int,Uua))) ).

% ATP.lambda_73
tff(fact_10283_ATP_Olambda__74,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_id,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(sgn_sgn_real,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(abs_abs_real,Uua)),Uu2)) ).

% ATP.lambda_74
tff(fact_10284_ATP_Olambda__75,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ep,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(sin_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_75
tff(fact_10285_ATP_Olambda__76,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_eo,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(cos_coeff,Uua)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_76
tff(fact_10286_ATP_Olambda__77,axiom,
    ! [Uu2: num,Uua: num] : aa_num_int(aa_num_fun_num_int(aTP_Lamm_lh,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_num_int(numeral_numeral_int,Uu2)),aa_num_int(numeral_numeral_int,Uua)) ).

% ATP.lambda_77
tff(fact_10287_ATP_Olambda__78,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_ho,Uu2),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu2),
        $let(
          l3: num,
          l3:= aa_num_num(plus_plus_num(l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_num_num(plus_plus_num(l3),one)) ) ) ).

% ATP.lambda_78
tff(fact_10288_ATP_Olambda__79,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_ky,Uu2),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu2),
        $let(
          l3: nat,
          l3:= aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l2),l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,l3),one_one_nat)) ) ) ).

% ATP.lambda_79
tff(fact_10289_ATP_Olambda__80,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_bu(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(semiri8010041392384452111omplex,Uua)),numera6690914467698888265omplex(aa_num_num(bit0,one)))) ).

% ATP.lambda_80
tff(fact_10290_ATP_Olambda__81,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_bv,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(semiri5074537144036343181t_real,Uua)),aa_num_real(numeral_numeral_real,aa_num_num(bit0,one)))) ).

% ATP.lambda_81
tff(fact_10291_ATP_Olambda__82,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_bw(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_rat_rat(divide_divide_rat(aa_nat_rat(semiri681578069525770553at_rat,Uua)),aa_num_rat(numeral_numeral_rat,aa_num_num(bit0,one)))) ).

% ATP.lambda_82
tff(fact_10292_ATP_Olambda__83,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fi,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uua)) ).

% ATP.lambda_83
tff(fact_10293_ATP_Olambda__84,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jl,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,one_one_real),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))))) ).

% ATP.lambda_84
tff(fact_10294_ATP_Olambda__85,axiom,
    ! [Uu2: nat,Uua: vEBT_VEBT] : aa_VEB5832953974749077948T_VEBT(aTP_Lamm_ma(Uu2),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))))) ).

% ATP.lambda_85
tff(fact_10295_ATP_Olambda__86,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_lt,Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_o_int(zero_n2684676970156552555ol_int,Uua != zero_zero_int)) ).

% ATP.lambda_86
tff(fact_10296_ATP_Olambda__87,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jj,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(uminus_uminus_real,aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua))))) ).

% ATP.lambda_87
tff(fact_10297_ATP_Olambda__88,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jg,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_real_real(inverse_inverse_real,aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uua)))) ).

% ATP.lambda_88
tff(fact_10298_ATP_Olambda__89,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ke(Uu2),Uua) = groups6591440286371151544t_real(Uu2,aa_nat_set_nat(set_ord_lessThan_nat,Uua)) ).

% ATP.lambda_89
tff(fact_10299_ATP_Olambda__90,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_db(Uu2),Uua) = aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_90
tff(fact_10300_ATP_Olambda__91,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dd,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_91
tff(fact_10301_ATP_Olambda__92,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_dc(Uu2),Uua) = aa_rat_rat(minus_minus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_92
tff(fact_10302_ATP_Olambda__93,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_de,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_93
tff(fact_10303_ATP_Olambda__94,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_df(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(minus_minus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_94
tff(fact_10304_ATP_Olambda__95,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat_complex(aTP_Lamm_cn(Uu2),Uua) = aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua)) ).

% ATP.lambda_95
tff(fact_10305_ATP_Olambda__96,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_co,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua)) ).

% ATP.lambda_96
tff(fact_10306_ATP_Olambda__97,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_cp(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua)) ).

% ATP.lambda_97
tff(fact_10307_ATP_Olambda__98,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cq,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua)) ).

% ATP.lambda_98
tff(fact_10308_ATP_Olambda__99,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_int(aTP_Lamm_cr(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua)) ).

% ATP.lambda_99
tff(fact_10309_ATP_Olambda__100,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_list_nat(aa_nat6157064778417537467st_nat(aTP_Lamm_lu,Uu2),Uua) = cons_nat(Uu2,nat_list_decode(Uua)) ).

% ATP.lambda_100
tff(fact_10310_ATP_Olambda__101,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ks,Uu2),Uua)
    <=> ( Uua = aa_nat_nat(suc,Uu2) ) ) ).

% ATP.lambda_101
tff(fact_10311_ATP_Olambda__102,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_eh,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uu2) ) ).

% ATP.lambda_102
tff(fact_10312_ATP_Olambda__103,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ic,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,Uua),Uu2) ).

% ATP.lambda_103
tff(fact_10313_ATP_Olambda__104,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ei,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu2) ) ).

% ATP.lambda_104
tff(fact_10314_ATP_Olambda__105,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_au,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uua),Uu2) ).

% ATP.lambda_105
tff(fact_10315_ATP_Olambda__106,axiom,
    ! [Uu2: rat,Uua: rat] : aa_rat_rat(aTP_Lamm_av(Uu2),Uua) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Uua),Uu2) ).

% ATP.lambda_106
tff(fact_10316_ATP_Olambda__107,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_aw,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uu2) ).

% ATP.lambda_107
tff(fact_10317_ATP_Olambda__108,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_ax(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uu2) ).

% ATP.lambda_108
tff(fact_10318_ATP_Olambda__109,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lp,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uu2) ).

% ATP.lambda_109
tff(fact_10319_ATP_Olambda__110,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hc,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uu2) ).

% ATP.lambda_110
tff(fact_10320_ATP_Olambda__111,axiom,
    ! [Uu2: nat,Uua: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_if,Uu2),Uua) = aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uu2) ).

% ATP.lambda_111
tff(fact_10321_ATP_Olambda__112,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_lq,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uua),Uu2) ).

% ATP.lambda_112
tff(fact_10322_ATP_Olambda__113,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hy,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu2) ).

% ATP.lambda_113
tff(fact_10323_ATP_Olambda__114,axiom,
    ! [Uu2: int,Uua: int] : aa_int_int(aTP_Lamm_hj(Uu2),Uua) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),Uu2) ).

% ATP.lambda_114
tff(fact_10324_ATP_Olambda__115,axiom,
    ! [Uu2: real,Uua: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ih,Uu2),Uua) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,Uua),Uu2) ).

% ATP.lambda_115
tff(fact_10325_ATP_Olambda__116,axiom,
    ! [Uu2: code_integer,Uua: code_integer] :
      ( aa_Code_integer_o(aa_Cod326357343816142457eger_o(aTP_Lamm_aj,Uu2),Uua)
    <=> aa_Code_integer_o(aa_Cod326357343816142457eger_o(dvd_dvd_Code_integer,Uua),Uu2) ) ).

% ATP.lambda_116
tff(fact_10326_ATP_Olambda__117,axiom,
    ! [Uu2: complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_af(Uu2),Uua)
    <=> aa_complex_o(dvd_dvd_complex(Uua),Uu2) ) ).

% ATP.lambda_117
tff(fact_10327_ATP_Olambda__118,axiom,
    ! [Uu2: real,Uua: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_ag,Uu2),Uua)
    <=> aa_real_o(aa_real_fun_real_o(dvd_dvd_real,Uua),Uu2) ) ).

% ATP.lambda_118
tff(fact_10328_ATP_Olambda__119,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ah,Uu2),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uua),Uu2) ) ).

% ATP.lambda_119
tff(fact_10329_ATP_Olambda__120,axiom,
    ! [Uu2: int,Uua: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ai,Uu2),Uua)
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,Uua),Uu2) ) ).

% ATP.lambda_120
tff(fact_10330_ATP_Olambda__121,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ey,Uu2),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),Uu2) ).

% ATP.lambda_121
tff(fact_10331_ATP_Olambda__122,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_je,Uu2),Uua) = aa_real_real(aa_nat_fun_real_real(root,Uua),Uu2) ).

% ATP.lambda_122
tff(fact_10332_ATP_Olambda__123,axiom,
    ! [Uu2: set_Extended_enat,Uua: extended_enat] :
      ( aa_Extended_enat_o(aTP_Lamm_a(Uu2),Uua)
    <=> member_Extended_enat(Uua,Uu2) ) ).

% ATP.lambda_123
tff(fact_10333_ATP_Olambda__124,axiom,
    ! [Uu2: set_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_aa(Uu2),Uua)
    <=> member_complex(Uua,Uu2) ) ).

% ATP.lambda_124
tff(fact_10334_ATP_Olambda__125,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_ab(Uu2),Uua)
    <=> member_real(Uua,Uu2) ) ).

% ATP.lambda_125
tff(fact_10335_ATP_Olambda__126,axiom,
    ! [Uu2: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ac(Uu2),Uua)
    <=> member_list_nat(Uua,Uu2) ) ).

% ATP.lambda_126
tff(fact_10336_ATP_Olambda__127,axiom,
    ! [Uu2: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ad(Uu2),Uua)
    <=> member_nat(Uua,Uu2) ) ).

% ATP.lambda_127
tff(fact_10337_ATP_Olambda__128,axiom,
    ! [Uu2: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_ae(Uu2),Uua)
    <=> member_int(Uua,Uu2) ) ).

% ATP.lambda_128
tff(fact_10338_ATP_Olambda__129,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_ev(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)) ).

% ATP.lambda_129
tff(fact_10339_ATP_Olambda__130,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_eu(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)) ).

% ATP.lambda_130
tff(fact_10340_ATP_Olambda__131,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_ie(Uu2),Uua) = aa_real_real(Uu2,aa_real_real(uminus_uminus_real,Uua)) ).

% ATP.lambda_131
tff(fact_10341_ATP_Olambda__132,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_cg(Uu2),Uua) = aa_nat_real(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_132
tff(fact_10342_ATP_Olambda__133,axiom,
    ! [Uu2: fun_nat_o,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_jr(Uu2),Uua)
    <=> aa_nat_o(Uu2,aa_nat_nat(suc,Uua)) ) ).

% ATP.lambda_133
tff(fact_10343_ATP_Olambda__134,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ch(Uu2),Uua) = aa_nat_rat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_134
tff(fact_10344_ATP_Olambda__135,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_nat(aTP_Lamm_ca(Uu2),Uua) = aa_nat_nat(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_135
tff(fact_10345_ATP_Olambda__136,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_int(aTP_Lamm_cb(Uu2),Uua) = aa_nat_int(Uu2,aa_nat_nat(suc,Uua)) ).

% ATP.lambda_136
tff(fact_10346_ATP_Olambda__137,axiom,
    ! [Uu2: nat,Uua: num] : aa_num_option_num(aTP_Lamm_hl(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uu2,Uua))) ).

% ATP.lambda_137
tff(fact_10347_ATP_Olambda__138,axiom,
    ! [Uu2: num,Uua: nat] : aa_nat_option_num(aTP_Lamm_hi(Uu2),Uua) = some_num(case_option_num_num(one,bit1,bit_take_bit_num(Uua,Uu2))) ).

% ATP.lambda_138
tff(fact_10348_ATP_Olambda__139,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_lo,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ln(Uu2),Uua)) ).

% ATP.lambda_139
tff(fact_10349_ATP_Olambda__140,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_gz,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gy(Uu2),Uua)) ).

% ATP.lambda_140
tff(fact_10350_ATP_Olambda__141,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_gx,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gw(Uu2),Uua)) ).

% ATP.lambda_141
tff(fact_10351_ATP_Olambda__142,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_gv,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_gu(Uu2),Uua)) ).

% ATP.lambda_142
tff(fact_10352_ATP_Olambda__143,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7695548938202568488_nat_o(aa_nat4259715169212259317_nat_o(aTP_Lamm_gt,Uu2),Uua) = produc6081775807080527818_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_gs(Uu2),Uua)) ).

% ATP.lambda_143
tff(fact_10353_ATP_Olambda__144,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat7997280740862404369at_nat(aa_nat5682243440470347102at_nat(aTP_Lamm_gr,Uu2),Uua) = produc2626176000494625587at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gq(Uu2),Uua)) ).

% ATP.lambda_144
tff(fact_10354_ATP_Olambda__145,axiom,
    ! [Uu2: fun_nat_real,Uua: real] : aa_real_real(aTP_Lamm_io(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(aTP_Lamm_in(Uu2),Uua)) ).

% ATP.lambda_145
tff(fact_10355_ATP_Olambda__146,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_complex(aTP_Lamm_ex(Uu2),Uua) = aa_real_complex(cis,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi)),aa_nat_real(semiri5074537144036343181t_real,Uua))),aa_nat_real(semiri5074537144036343181t_real,Uu2))) ).

% ATP.lambda_146
tff(fact_10356_ATP_Olambda__147,axiom,
    ! [Uu2: fun_real_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_jw(Uu2),Uua)
    <=> ( aa_real_real(Uu2,Uua) != zero_zero_real ) ) ).

% ATP.lambda_147
tff(fact_10357_ATP_Olambda__148,axiom,
    ! [Uu2: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bk,Uu2),Uua)
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua))) ) ).

% ATP.lambda_148
tff(fact_10358_ATP_Olambda__149,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_bq(Uu2),Uua) = times_times_complex(aa_complex_complex(minus_minus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_149
tff(fact_10359_ATP_Olambda__150,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bs(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_150
tff(fact_10360_ATP_Olambda__151,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_br(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(minus_minus_rat(Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_151
tff(fact_10361_ATP_Olambda__152,axiom,
    ! [Uu2: complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_bl(Uu2),Uua) = times_times_complex(aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,Uua))) ).

% ATP.lambda_152
tff(fact_10362_ATP_Olambda__153,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_bm(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,Uua))) ).

% ATP.lambda_153
tff(fact_10363_ATP_Olambda__154,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_bn(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,Uua))) ).

% ATP.lambda_154
tff(fact_10364_ATP_Olambda__155,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_bp(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,Uua))) ).

% ATP.lambda_155
tff(fact_10365_ATP_Olambda__156,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_bo(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,Uua))) ).

% ATP.lambda_156
tff(fact_10366_ATP_Olambda__157,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_lx(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_157
tff(fact_10367_ATP_Olambda__158,axiom,
    ! [Uu2: real,Uua: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ji,Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uua)) ).

% ATP.lambda_158
tff(fact_10368_ATP_Olambda__159,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_mf(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uua)) ).

% ATP.lambda_159
tff(fact_10369_ATP_Olambda__160,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_md(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uua)) ).

% ATP.lambda_160
tff(fact_10370_ATP_Olambda__161,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_Extended_enat(aTP_Lamm_mb(Uu2),Uua) = extended_enat2(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uua)) ).

% ATP.lambda_161
tff(fact_10371_ATP_Olambda__162,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hp,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uu2),Uua)) ).

% ATP.lambda_162
tff(fact_10372_ATP_Olambda__163,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_hq,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_min_nat,Uua),Uu2)) ).

% ATP.lambda_163
tff(fact_10373_ATP_Olambda__164,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gc,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uu2),Uua)) ).

% ATP.lambda_164
tff(fact_10374_ATP_Olambda__165,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_gb,Uu2),Uua) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Uua),Uu2)) ).

% ATP.lambda_165
tff(fact_10375_ATP_Olambda__166,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jc(Uu2),Uua) = aa_real_real(inverse_inverse_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_166
tff(fact_10376_ATP_Olambda__167,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aTP_Lamm_kz(Uu2),Uua) = aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_167
tff(fact_10377_ATP_Olambda__168,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kv,Uu2),Uua) = aa_rat_rat(uminus_uminus_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_168
tff(fact_10378_ATP_Olambda__169,axiom,
    ! [Uu2: fun_nat_complex,Uua: nat] : aa_nat3829984526574172007omplex(aTP_Lamm_ci(Uu2),Uua) = times_times_complex(aa_nat_complex(Uu2,Uua)) ).

% ATP.lambda_169
tff(fact_10379_ATP_Olambda__170,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_fun_real_real(aTP_Lamm_cj(Uu2),Uua) = aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_170
tff(fact_10380_ATP_Olambda__171,axiom,
    ! [Uu2: fun_nat_rat,Uua: nat] : aa_nat_fun_rat_rat(aTP_Lamm_ck(Uu2),Uua) = aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uua)) ).

% ATP.lambda_171
tff(fact_10381_ATP_Olambda__172,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat] : aa_nat_fun_nat_nat(aTP_Lamm_cl(Uu2),Uua) = aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uua)) ).

% ATP.lambda_172
tff(fact_10382_ATP_Olambda__173,axiom,
    ! [Uu2: fun_nat_int,Uua: nat] : aa_nat_fun_int_int(aTP_Lamm_cm(Uu2),Uua) = aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uua)) ).

% ATP.lambda_173
tff(fact_10383_ATP_Olambda__174,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_kg(Uu2),Uua) = aa_real_real(artanh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_174
tff(fact_10384_ATP_Olambda__175,axiom,
    ! [Uu2: fun_real_real,Uua: real] : aa_real_real(aTP_Lamm_kf(Uu2),Uua) = aa_real_real(arcosh_real,aa_real_real(Uu2,Uua)) ).

% ATP.lambda_175
tff(fact_10385_ATP_Olambda__176,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_dy(Uu2),Uua) = aa_real_real(abs_abs_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_176
tff(fact_10386_ATP_Olambda__177,axiom,
    ! [Uu2: fun_nat_real,Uua: nat] : aa_nat_real(aTP_Lamm_jo(Uu2),Uua) = aa_real_real(cos_real,aa_nat_real(Uu2,Uua)) ).

% ATP.lambda_177
tff(fact_10387_ATP_Olambda__178,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: real] : aa_real_real(aTP_Lamm_il(Uu2),Uua) = suminf_real(aa_real_fun_nat_real(Uu2,Uua)) ).

% ATP.lambda_178
tff(fact_10388_ATP_Olambda__179,axiom,
    ! [Uu2: list_VEBT_VEBT,Uua: vEBT_VEBT] :
      ( aa_VEBT_VEBT_o(aTP_Lamm_bg(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = nth_VEBT_VEBT(Uu2,I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6755466524823107622T_VEBT(Uu2)) ) ) ).

% ATP.lambda_179
tff(fact_10389_ATP_Olambda__180,axiom,
    ! [Uu2: list_complex,Uua: complex] :
      ( aa_complex_o(aTP_Lamm_bd(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_complex(nth_complex(Uu2),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3451745648224563538omplex(Uu2)) ) ) ).

% ATP.lambda_180
tff(fact_10390_ATP_Olambda__181,axiom,
    ! [Uu2: list_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_be(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_real(nth_real(Uu2),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_real(Uu2)) ) ) ).

% ATP.lambda_181
tff(fact_10391_ATP_Olambda__182,axiom,
    ! [Uu2: list_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_bf(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = nth_list_nat(Uu2,I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3023201423986296836st_nat(Uu2)) ) ) ).

% ATP.lambda_182
tff(fact_10392_ATP_Olambda__183,axiom,
    ! [Uu2: list_o,Uua: $o] :
      ( aa_o_o(aTP_Lamm_bh(Uu2),(Uua))
    <=> ? [I2: nat] :
          ( ( (Uua)
          <=> aa_nat_o(nth_o(Uu2),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_o(Uu2)) ) ) ).

% ATP.lambda_183
tff(fact_10393_ATP_Olambda__184,axiom,
    ! [Uu2: list_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_bi(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_nat(nth_nat(Uu2),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Uu2)) ) ) ).

% ATP.lambda_184
tff(fact_10394_ATP_Olambda__185,axiom,
    ! [Uu2: list_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_bj(Uu2),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa_nat_int(nth_int(Uu2),I2) )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Uu2)) ) ) ).

% ATP.lambda_185
tff(fact_10395_ATP_Olambda__186,axiom,
    ! [Uu2: set_real,Uua: real] :
      ( aa_real_o(aTP_Lamm_lm(Uu2),Uua)
    <=> ! [X: real] :
          ( member_real(X,Uu2)
         => aa_real_o(aa_real_fun_real_o(ord_less_eq_real,X),Uua) ) ) ).

% ATP.lambda_186
tff(fact_10396_ATP_Olambda__187,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_ly(Uu2),Uua) = aa_nat_set_nat(set_ord_atMost_nat,Uu2) ).

% ATP.lambda_187
tff(fact_10397_ATP_Olambda__188,axiom,
    ! [Uu2: nat,Uua: nat] : aa_nat_rat(aTP_Lamm_lc(Uu2),Uua) = aa_nat_rat(semiri681578069525770553at_rat,Uu2) ).

% ATP.lambda_188
tff(fact_10398_ATP_Olambda__189,axiom,
    ! [Uu2: int,Uua: nat] : aa_nat_rat(aTP_Lamm_le(Uu2),Uua) = aa_int_rat(ring_1_of_int_rat,Uu2) ).

% ATP.lambda_189
tff(fact_10399_ATP_Olambda__190,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_ed(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uua,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,Uub),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))),aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),one_one_nat)),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))))) ).

% ATP.lambda_190
tff(fact_10400_ATP_Olambda__191,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] :
      aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_et(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uub),aa_nat_real(Uu2,Uub),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_191
tff(fact_10401_ATP_Olambda__192,axiom,
    ! [Uu2: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_ft(Uu2),Uua),Uub) = $ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le3102999989581377725nteger,numera6620942414471956472nteger(Uu2)),Uub),produc1086072967326762835nteger(plus_p5714425477246183910nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu2))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_192
tff(fact_10402_ATP_Olambda__193,axiom,
    ! [Uu2: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_fl(Uu2),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_num_nat(numeral_numeral_nat,Uu2)),Uub),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),aa_num_nat(numeral_numeral_nat,Uu2))),aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uua)),Uub)) ).

% ATP.lambda_193
tff(fact_10403_ATP_Olambda__194,axiom,
    ! [Uu2: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_fm(Uu2),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_num_int(numeral_numeral_int,Uu2)),Uub),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),aa_num_int(numeral_numeral_int,Uu2))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,aa_num_int(numeral_numeral_int,aa_num_num(bit0,one))),Uua),Uub)) ).

% ATP.lambda_194
tff(fact_10404_ATP_Olambda__195,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fy(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu2),Uub))) ).

% ATP.lambda_195
tff(fact_10405_ATP_Olambda__196,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_go(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu2),Uub))) ).

% ATP.lambda_196
tff(fact_10406_ATP_Olambda__197,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_fx(Uu2),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu2,Uub))) ).

% ATP.lambda_197
tff(fact_10407_ATP_Olambda__198,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] :
      ( aa_real_o(aa_fun2895569901277098738real_o(aTP_Lamm_jt(Uu2),Uua),Uub)
    <=> has_fi5821293074295781190e_real(Uu2,aa_real_real(Uua,Uub),topolo2177554685111907308n_real(Uub,top_top_set_real)) ) ).

% ATP.lambda_198
tff(fact_10408_ATP_Olambda__199,axiom,
    ! [Uu2: fun_re4238438383274954274t_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ik(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(Uu2,Uub),Uua) ).

% ATP.lambda_199
tff(fact_10409_ATP_Olambda__200,axiom,
    ! [Uu2: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_fw(Uu2),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(aa_Cod326357343816142457eger_o(ord_le6747313008572928689nteger,zero_z3403309356797280102nteger),Uu2),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_200
tff(fact_10410_ATP_Olambda__201,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kk(Uu2),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kj(Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_201
tff(fact_10411_ATP_Olambda__202,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ki(Uu2),Uua),Uub)
    <=> aa_Pro4592068669816069595_int_o(produc4947309494688390418_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kh(Uua),Uub)),quotient_of(Uu2)) ) ).

% ATP.lambda_202
tff(fact_10412_ATP_Olambda__203,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gn(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gm(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_203
tff(fact_10413_ATP_Olambda__204,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gl(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gk(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_204
tff(fact_10414_ATP_Olambda__205,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gj(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gi(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_205
tff(fact_10415_ATP_Olambda__206,axiom,
    ! [Uu2: rat,Uua: int,Uub: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_gh(Uu2),Uua),Uub) = produc4245557441103728435nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gg(Uua),Uub),quotient_of(Uu2)) ).

% ATP.lambda_206
tff(fact_10416_ATP_Olambda__207,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_ip(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_207
tff(fact_10417_ATP_Olambda__208,axiom,
    ! [Uu2: real,Uua: fun_na6724076748692969250l_real,Uub: nat] : aa_nat_real(aa_fun1113820866546176217t_real(aTP_Lamm_iq(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uua,Uub),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_208
tff(fact_10418_ATP_Olambda__209,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_im(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(suc,Uub)))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_209
tff(fact_10419_ATP_Olambda__210,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_es(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uua,Uub)),aa_nat_real(semiri2265585572941072030t_real,Uub))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_210
tff(fact_10420_ATP_Olambda__211,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hv(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_211
tff(fact_10421_ATP_Olambda__212,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hw(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_212
tff(fact_10422_ATP_Olambda__213,axiom,
    ! [Uu2: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_hu(Uu2),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu2 )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_213
tff(fact_10423_ATP_Olambda__214,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fu(Uu2),Uua),Uub)
    <=> ( member_nat(aa_nat_nat(suc,Uub),Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_214
tff(fact_10424_ATP_Olambda__215,axiom,
    ! [Uu2: set_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_fv(Uu2),Uua),Uub)
    <=> ( member_nat(Uub,Uu2)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),aa_nat_nat(suc,Uua)) ) ) ).

% ATP.lambda_215
tff(fact_10425_ATP_Olambda__216,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fb(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uub)) ).

% ATP.lambda_216
tff(fact_10426_ATP_Olambda__217,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_km(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uua)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_217
tff(fact_10427_ATP_Olambda__218,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_kl(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uua),Uub) ) ) ).

% ATP.lambda_218
tff(fact_10428_ATP_Olambda__219,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_el(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_219
tff(fact_10429_ATP_Olambda__220,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ek(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Uub),Uua) ) ) ).

% ATP.lambda_220
tff(fact_10430_ATP_Olambda__221,axiom,
    ! [Uu2: int,Uua: int,Uub: int] :
      ( aa_int_o(aa_int_fun_int_o(aTP_Lamm_ej(Uu2),Uua),Uub)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Uu2),Uub)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Uub),Uua) ) ) ).

% ATP.lambda_221
tff(fact_10431_ATP_Olambda__222,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hs(Uu2),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uua)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,Uub),Uu2) ) ) ).

% ATP.lambda_222
tff(fact_10432_ATP_Olambda__223,axiom,
    ! [Uu2: set_Extended_enat,Uua: set_Extended_enat,Uub: extended_enat] :
      ( aa_Extended_enat_o(aa_set3572156908926473983enat_o(aTP_Lamm_ak(Uu2),Uua),Uub)
    <=> ( member_Extended_enat(Uub,Uu2)
        & ~ member_Extended_enat(Uub,Uua) ) ) ).

% ATP.lambda_223
tff(fact_10433_ATP_Olambda__224,axiom,
    ! [Uu2: set_complex,Uua: set_complex,Uub: complex] :
      ( aa_complex_o(aa_set1742385355640705825plex_o(aTP_Lamm_al(Uu2),Uua),Uub)
    <=> ( member_complex(Uub,Uu2)
        & ~ member_complex(Uub,Uua) ) ) ).

% ATP.lambda_224
tff(fact_10434_ATP_Olambda__225,axiom,
    ! [Uu2: set_real,Uua: set_real,Uub: real] :
      ( aa_real_o(aa_set2323979645938365349real_o(aTP_Lamm_am(Uu2),Uua),Uub)
    <=> ( member_real(Uub,Uu2)
        & ~ member_real(Uub,Uua) ) ) ).

% ATP.lambda_225
tff(fact_10435_ATP_Olambda__226,axiom,
    ! [Uu2: set_list_nat,Uua: set_list_nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_set6675653781276780861_nat_o(aTP_Lamm_an(Uu2),Uua),Uub)
    <=> ( member_list_nat(Uub,Uu2)
        & ~ member_list_nat(Uub,Uua) ) ) ).

% ATP.lambda_226
tff(fact_10436_ATP_Olambda__227,axiom,
    ! [Uu2: set_nat,Uua: set_nat,Uub: nat] :
      ( aa_nat_o(aa_set_nat_fun_nat_o(aTP_Lamm_ap(Uu2),Uua),Uub)
    <=> ( member_nat(Uub,Uu2)
        & ~ member_nat(Uub,Uua) ) ) ).

% ATP.lambda_227
tff(fact_10437_ATP_Olambda__228,axiom,
    ! [Uu2: set_int,Uua: set_int,Uub: int] :
      ( aa_int_o(aa_set_int_fun_int_o(aTP_Lamm_ao(Uu2),Uua),Uub)
    <=> ( member_int(Uub,Uu2)
        & ~ member_int(Uub,Uua) ) ) ).

% ATP.lambda_228
tff(fact_10438_ATP_Olambda__229,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_hr(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)),Uu2) ) ).

% ATP.lambda_229
tff(fact_10439_ATP_Olambda__230,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_lw(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)),Uu2) ) ).

% ATP.lambda_230
tff(fact_10440_ATP_Olambda__231,axiom,
    ! [Uu2: real,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_kp(Uu2),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_complex_real(real_V3694042436643373181omplex(Uua),Uub)),Uu2) ) ).

% ATP.lambda_231
tff(fact_10441_ATP_Olambda__232,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kn(Uu2),Uua),Uub)
    <=> aa_real_o(aa_real_fun_real_o(ord_less_real,aa_real_real(aa_rea7340605660300375871l_real(real_V975177566351809787t_real,Uua),Uub)),Uu2) ) ).

% ATP.lambda_232
tff(fact_10442_ATP_Olambda__233,axiom,
    ! [Uu2: nat,Uua: complex,Uub: complex] :
      ( aa_complex_o(aa_com5808801942499186519plex_o(aTP_Lamm_em(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uu2) = Uua ) ) ).

% ATP.lambda_233
tff(fact_10443_ATP_Olambda__234,axiom,
    ! [Uu2: complex,Uua: nat,Uub: complex] :
      ( aa_complex_o(aa_nat_fun_complex_o(aTP_Lamm_fj(Uu2),Uua),Uub)
    <=> ( aa_nat_complex(power_power_complex(Uub),Uua) = Uu2 ) ) ).

% ATP.lambda_234
tff(fact_10444_ATP_Olambda__235,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_int,Uub: nat] : aa_nat_real(aa_fun4551936527957760560t_real(aTP_Lamm_jn(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_int_real(ring_1_of_int_real,aa_nat_int(Uua,Uub))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_num_real(numeral_numeral_real,aa_num_num(bit0,one))),pi))) ).

% ATP.lambda_235
tff(fact_10445_ATP_Olambda__236,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_in(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),aa_nat_nat(suc,Uub))) ).

% ATP.lambda_236
tff(fact_10446_ATP_Olambda__237,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dz(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_nat_real(Uu2,Uub)),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uua),Uub)) ).

% ATP.lambda_237
tff(fact_10447_ATP_Olambda__238,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_fe(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),Uub)) ).

% ATP.lambda_238
tff(fact_10448_ATP_Olambda__239,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_ef(Uu2),Uua),Uub)
    <=> ( aa_nat_o(Uu2,Uub)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uua) ) ) ).

% ATP.lambda_239
tff(fact_10449_ATP_Olambda__240,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ju(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_240
tff(fact_10450_ATP_Olambda__241,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_jx(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(Uua,Uub)),aa_real_real(Uu2,Uub)) ).

% ATP.lambda_241
tff(fact_10451_ATP_Olambda__242,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kt,Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_242
tff(fact_10452_ATP_Olambda__243,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat] : aa_nat_nat(aa_fun5791449513213591928at_nat(aTP_Lamm_dj(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uub)),aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_243
tff(fact_10453_ATP_Olambda__244,axiom,
    ! [Uu2: fun_nat_int,Uua: fun_nat_int,Uub: nat] : aa_nat_int(aa_fun5565043122120104496at_int(aTP_Lamm_dk(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(Uu2,Uub)),aa_nat_int(Uua,Uub)) ).

% ATP.lambda_244
tff(fact_10454_ATP_Olambda__245,axiom,
    ! [Uu2: fun_int_int,Uua: fun_int_int,Uub: int] : aa_int_int(aa_fun6280345759651049192nt_int(aTP_Lamm_dl(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(Uu2,Uub)),aa_int_int(Uua,Uub)) ).

% ATP.lambda_245
tff(fact_10455_ATP_Olambda__246,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_jb(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_246
tff(fact_10456_ATP_Olambda__247,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_kx(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_247
tff(fact_10457_ATP_Olambda__248,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aa_fun3638354034882855594at_rat(aTP_Lamm_kw,Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_nat_rat(Uu2,Uub)),aa_nat_rat(Uua,Uub)) ).

% ATP.lambda_248
tff(fact_10458_ATP_Olambda__249,axiom,
    ! [Uu2: fun_real_real,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_ij(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),aa_real_real(Uua,Uub)) ).

% ATP.lambda_249
tff(fact_10459_ATP_Olambda__250,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_eg(Uu2),Uua),Uub)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(Uu2,Uub)),Uua) ) ).

% ATP.lambda_250
tff(fact_10460_ATP_Olambda__251,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_bx(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_251
tff(fact_10461_ATP_Olambda__252,axiom,
    ! [Uu2: fun_nat_int,Uua: int,Uub: nat] : aa_nat_int(aa_int_fun_nat_int(aTP_Lamm_by(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_252
tff(fact_10462_ATP_Olambda__253,axiom,
    ! [Uu2: fun_int_int,Uua: int,Uub: int] : aa_int_int(aa_int_fun_int_int(aTP_Lamm_bz(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_253
tff(fact_10463_ATP_Olambda__254,axiom,
    ! [Uu2: fun_nat_complex,Uua: complex,Uub: nat] : aa_nat_complex(aa_com8181001469867462503omplex(aTP_Lamm_dw(Uu2),Uua),Uub) = aa_complex_complex(divide1717551699836669952omplex(aa_nat_complex(Uu2,Uub)),Uua) ).

% ATP.lambda_254
tff(fact_10464_ATP_Olambda__255,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_dx(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(Uu2,Uub)),Uua) ).

% ATP.lambda_255
tff(fact_10465_ATP_Olambda__256,axiom,
    ! [Uu2: fun_real_real,Uua: nat,Uub: real] : aa_real_real(aa_nat_fun_real_real(aTP_Lamm_ig(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_256
tff(fact_10466_ATP_Olambda__257,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_dm(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(Uu2,Uub)),Uua) ).

% ATP.lambda_257
tff(fact_10467_ATP_Olambda__258,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_dn(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_nat_int(Uu2,Uub)),Uua) ).

% ATP.lambda_258
tff(fact_10468_ATP_Olambda__259,axiom,
    ! [Uu2: fun_int_int,Uua: nat,Uub: int] : aa_int_int(aa_nat_fun_int_int(aTP_Lamm_do(Uu2),Uua),Uub) = aa_nat_int(power_power_int(aa_int_int(Uu2,Uub)),Uua) ).

% ATP.lambda_259
tff(fact_10469_ATP_Olambda__260,axiom,
    ! [Uu2: nat,Uua: fun_real_real,Uub: real] : aa_real_real(aa_fun6946558556880649704l_real(aTP_Lamm_kb(Uu2),Uua),Uub) = aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(Uua,Uub)),Uu2) ).

% ATP.lambda_260
tff(fact_10470_ATP_Olambda__261,axiom,
    ! [Uu2: fun_real_real,Uua: real,Uub: real] : aa_real_real(aa_rea7340605660300375871l_real(aTP_Lamm_ii(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real,aa_real_real(Uu2,Uub)),Uua) ).

% ATP.lambda_261
tff(fact_10471_ATP_Olambda__262,axiom,
    ! [Uu2: fun_nat_rat,Uua: fun_nat_rat,Uub: nat] : aa_nat_rat(aa_fun765758790203053448at_rat(aTP_Lamm_la(Uu2),Uua),Uub) = aa_rat_rat(minus_minus_rat(aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uu2,Uub))),aa_rat_rat(inverse_inverse_rat,aa_nat_rat(Uua,Uub))) ).

% ATP.lambda_262
tff(fact_10472_ATP_Olambda__263,axiom,
    ! [Uu2: real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jh(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,Uua),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uu2),Uub)) ).

% ATP.lambda_263
tff(fact_10473_ATP_Olambda__264,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_en(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uub),Uua)) ).

% ATP.lambda_264
tff(fact_10474_ATP_Olambda__265,axiom,
    ! [Uu2: real,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_jv(Uu2),Uua),Uub)
    <=> member_real(Uub,set_or1633881224788618240n_real(Uu2,Uua)) ) ).

% ATP.lambda_265
tff(fact_10475_ATP_Olambda__266,axiom,
    ! [Uu2: real,Uua: fun_nat_real,Uub: nat] : aa_nat_real(aa_fun1839206346889086768t_real(aTP_Lamm_dv(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uu2),aa_nat_real(Uua,Uub)) ).

% ATP.lambda_266
tff(fact_10476_ATP_Olambda__267,axiom,
    ! [Uu2: complex,Uua: nat,Uub: nat] : aa_nat_complex(aa_nat6454746962644518025omplex(aTP_Lamm_cs(Uu2),Uua),Uub) = aa_complex_complex(plus_plus_complex(Uu2),aa_nat_complex(semiri8010041392384452111omplex,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_267
tff(fact_10477_ATP_Olambda__268,axiom,
    ! [Uu2: real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_ct(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uu2),aa_nat_real(semiri5074537144036343181t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_268
tff(fact_10478_ATP_Olambda__269,axiom,
    ! [Uu2: rat,Uua: nat,Uub: nat] : aa_nat_rat(aa_nat_fun_nat_rat(aTP_Lamm_cu(Uu2),Uua),Uub) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Uu2),aa_nat_rat(semiri681578069525770553at_rat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_269
tff(fact_10479_ATP_Olambda__270,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cv(Uu2),Uua),Uub) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_270
tff(fact_10480_ATP_Olambda__271,axiom,
    ! [Uu2: int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_cw(Uu2),Uua),Uub) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Uu2),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uua),Uub))) ).

% ATP.lambda_271
tff(fact_10481_ATP_Olambda__272,axiom,
    ! [Uu2: real,Uua: real,Uub: product_unit] : aa_Product_unit_real(aa_rea7683167438103758690t_real(aTP_Lamm_lv(Uu2),Uua),Uub) = aa_real_real(aa_rea7340605660300375871l_real(powr_real2,Uu2),Uua) ).

% ATP.lambda_272
tff(fact_10482_ATP_Olambda__273,axiom,
    ! [Uu2: fun_real_o,Uua: real,Uub: real] :
      ( aa_real_o(aa_real_fun_real_o(aTP_Lamm_kd(Uu2),Uua),Uub)
    <=> aa_real_o(Uu2,aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,Uub),Uua)) ) ).

% ATP.lambda_273
tff(fact_10483_ATP_Olambda__274,axiom,
    ! [Uu2: fun_nat_real,Uua: nat,Uub: nat] : aa_nat_real(aa_nat_fun_nat_real(aTP_Lamm_du(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_274
tff(fact_10484_ATP_Olambda__275,axiom,
    ! [Uu2: fun_nat_o,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_js(Uu2),Uua),Uub)
    <=> aa_nat_o(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_275
tff(fact_10485_ATP_Olambda__276,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_cc(Uu2),Uua),Uub) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_276
tff(fact_10486_ATP_Olambda__277,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat] : aa_nat_int(aa_nat_fun_nat_int(aTP_Lamm_cd(Uu2),Uua),Uub) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ).

% ATP.lambda_277
tff(fact_10487_ATP_Olambda__278,axiom,
    ! [Uu2: fun_nat_real,Uua: fun_nat_nat,Uub: nat] : aa_nat_real(aa_fun1538629612289296340t_real(aTP_Lamm_kr(Uu2),Uua),Uub) = aa_nat_real(Uu2,aa_nat_nat(Uua,Uub)) ).

% ATP.lambda_278
tff(fact_10488_ATP_Olambda__279,axiom,
    ! [Uu2: fun_nat_real,Uua: real,Uub: nat] : aa_nat_real(aa_real_fun_nat_real(aTP_Lamm_jm(Uu2),Uua),Uub) = aa_real_real(cos_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_nat_real(Uu2,Uub)),Uua)) ).

% ATP.lambda_279
tff(fact_10489_ATP_Olambda__280,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_fg(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_set_nat_nat(groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ff(Uu2),Uua),Uuc)),aa_nat_set_nat(set_ord_atMost_nat,Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uub),Uuc)) ).

% ATP.lambda_280
tff(fact_10490_ATP_Olambda__281,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_iv(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(suc,Uua)),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_281
tff(fact_10491_ATP_Olambda__282,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: nat,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_it(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)),zero_zero_real)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,Uub),Uuc)) ).

% ATP.lambda_282
tff(fact_10492_ATP_Olambda__283,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_ir(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uua)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uub),Uua)),Uuc)) ).

% ATP.lambda_283
tff(fact_10493_ATP_Olambda__284,axiom,
    ! [Uu2: fun_na6724076748692969250l_real,Uua: real,Uub: real,Uuc: nat] : aa_nat_real(aa_real_fun_nat_real(aa_rea7841385751566875788t_real(aTP_Lamm_is(Uu2),Uua),Uub),Uuc) = aa_real_real(aa_rea7340605660300375871l_real(times_times_real,aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_real_real(aa_nat_fun_real_real(Uu2,Uuc),Uub)),aa_nat_real(semiri2265585572941072030t_real,Uuc))),aa_nat_real(aa_real_fun_nat_real(power_power_real,aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,Uua),Uub)),Uuc)) ).

% ATP.lambda_284
tff(fact_10494_ATP_Olambda__285,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gq(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Uua),Uub))) ).

% ATP.lambda_285
tff(fact_10495_ATP_Olambda__286,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_fd(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(semiri1316708129612266289at_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uub),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uu2),Uuc))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_286
tff(fact_10496_ATP_Olambda__287,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_fc(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(binomial,Uua),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_287
tff(fact_10497_ATP_Olambda__288,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kj(Uu2),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_288
tff(fact_10498_ATP_Olambda__289,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_gu(Uu2),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_289
tff(fact_10499_ATP_Olambda__290,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] :
      ( aa_int_o(aa_int_fun_int_o(aa_int2464330468366073224_int_o(aTP_Lamm_kh(Uu2),Uua),Uub),Uuc)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub)) ) ).

% ATP.lambda_290
tff(fact_10500_ATP_Olambda__291,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_gs(Uu2),Uua),Uub),Uuc)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)) ) ).

% ATP.lambda_291
tff(fact_10501_ATP_Olambda__292,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gw(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uub)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uuc)) ).

% ATP.lambda_292
tff(fact_10502_ATP_Olambda__293,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aa_nat4158579785191972357at_nat(aTP_Lamm_gy(Uu2),Uua),Uub),Uuc) = aa_nat7603221163087113075at_nat(product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uub)) ).

% ATP.lambda_293
tff(fact_10503_ATP_Olambda__294,axiom,
    ! [Uu2: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aTP_Lamm_ln(Uu2),Uua),Uub),Uuc)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uu2),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua) ) ) ).

% ATP.lambda_294
tff(fact_10504_ATP_Olambda__295,axiom,
    ! [Uu2: fun_nat_nat,Uua: fun_nat_nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_fun6650658898577385413at_nat(aTP_Lamm_ff(Uu2),Uua),Uub),Uuc) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(Uu2,Uuc)),aa_nat_nat(Uua,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),Uuc))) ).

% ATP.lambda_295
tff(fact_10505_ATP_Olambda__296,axiom,
    ! [Uu2: fun_nat_nat,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aa_nat3783148529270815480at_nat(aTP_Lamm_ce(Uu2),Uua),Uub),Uuc) = aa_nat_nat(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_296
tff(fact_10506_ATP_Olambda__297,axiom,
    ! [Uu2: fun_nat_int,Uua: nat,Uub: nat,Uuc: nat] : aa_nat_int(aa_nat_fun_nat_int(aa_nat1293065384339088724at_int(aTP_Lamm_cf(Uu2),Uua),Uub),Uuc) = aa_nat_int(Uu2,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uub),Uua)),Uuc)) ).

% ATP.lambda_297
tff(fact_10507_ATP_Olambda__298,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gm(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_298
tff(fact_10508_ATP_Olambda__299,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gk(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc)),aa_int_int(aa_int_fun_int_int(times_times_int,Uub),Uua)),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_299
tff(fact_10509_ATP_Olambda__300,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gi(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uub),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uuc))) ).

% ATP.lambda_300
tff(fact_10510_ATP_Olambda__301,axiom,
    ! [Uu2: int,Uua: int,Uub: int,Uuc: int] : aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aa_int1966403025003302569nt_int(aTP_Lamm_gg(Uu2),Uua),Uub),Uuc) = normalize(product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,Uu2),Uuc),aa_int_int(aa_int_fun_int_int(times_times_int,Uua),Uub))) ).

% ATP.lambda_301
tff(fact_10511_ATP_Olambda__302,axiom,
    ! [Uu2: nat,Uua: fun_na6724076748692969250l_real,Uub: real,Uuc: nat,Uud: real] : aa_real_real(aa_nat_fun_real_real(aa_rea1103652080130114956l_real(aa_fun2567993205388704171l_real(aTP_Lamm_iu(Uu2),Uua),Uub),Uuc),Uud) = aa_real_real(aa_rea7340605660300375871l_real(minus_minus_real,aa_real_real(aa_nat_fun_real_real(Uua,Uuc),Uud)),aa_real_real(aa_rea7340605660300375871l_real(plus_plus_real,groups6591440286371151544t_real(aa_real_fun_nat_real(aa_nat1308781354268537904t_real(aTP_Lamm_it(Uua),Uuc),Uud),aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))),aa_real_real(aa_rea7340605660300375871l_real(times_times_real,Uub),aa_real_real(aa_rea7340605660300375871l_real(divide_divide_real,aa_nat_real(aa_real_fun_nat_real(power_power_real,Uud),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc))),aa_nat_real(semiri2265585572941072030t_real,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu2),Uuc)))))) ).

% ATP.lambda_302
tff(fact_10512_ATP_Olambda__303,axiom,
    ! [Uu2: nat,Uua: list_VEBT_VEBT,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aa_nat8314798964848609180_nat_o(aa_nat7293937527460703909_nat_o(aa_VEB4248176028289774660_nat_o(aa_lis144135085560928841_nat_o(aTP_Lamm_hb(Uu2),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uue),Uuf)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uuf),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
        & ! [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))),Uud))
           => ( ? [X_13: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(nth_VEBT_VEBT(Uua,I2)),X_13)
            <=> aa_nat_o(vEBT_V8194947554948674370ptions(Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member_VEBT_VEBT(X,set_VEBT_VEBT2(Uua))
             => ~ ? [X6: nat] : aa_nat_o(vEBT_V8194947554948674370ptions(X),X6) ),
            ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),Uuf)
            & ! [X: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one))),Uu2))
               => ( aa_nat_o(vEBT_V5917875025757280293ildren(Uuc,Uua),X)
                 => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uue),X)
                    & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_303
tff(fact_10513_ATP_Olambda__304,axiom,
    ! [Uu2: rat,Uua: nat] : aa_nat_rat(aTP_Lamm_ku(Uu2),Uua) = Uu2 ).

% ATP.lambda_304
tff(fact_10514_ATP_Olambda__305,axiom,
    ! [Uu2: complex] : aa_complex_complex(aTP_Lamm_ay,Uu2) = Uu2 ).

% ATP.lambda_305
tff(fact_10515_ATP_Olambda__306,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_az,Uu2) = Uu2 ).

% ATP.lambda_306
tff(fact_10516_ATP_Olambda__307,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_ba,Uu2) = Uu2 ).

% ATP.lambda_307
tff(fact_10517_ATP_Olambda__308,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_bb,Uu2) = Uu2 ).

% ATP.lambda_308
tff(fact_10518_ATP_Olambda__309,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_bc,Uu2) = Uu2 ).

% ATP.lambda_309
tff(fact_10519_ATP_Olambda__310,axiom,
    ! [Uu2: real] : aa_real_real(aTP_Lamm_aq,Uu2) = zero_zero_real ).

% ATP.lambda_310
tff(fact_10520_ATP_Olambda__311,axiom,
    ! [Uu2: rat] : aa_rat_rat(aTP_Lamm_ar,Uu2) = zero_zero_rat ).

% ATP.lambda_311
tff(fact_10521_ATP_Olambda__312,axiom,
    ! [Uu2: nat] : aa_nat_rat(aTP_Lamm_ld,Uu2) = zero_zero_rat ).

% ATP.lambda_312
tff(fact_10522_ATP_Olambda__313,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_as,Uu2) = zero_zero_nat ).

% ATP.lambda_313
tff(fact_10523_ATP_Olambda__314,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_at,Uu2) = zero_zero_int ).

% ATP.lambda_314
tff(fact_10524_ATP_Olambda__315,axiom,
    ! [Uu2: nat] : aa_nat_rat(aTP_Lamm_lb,Uu2) = one_one_rat ).

% ATP.lambda_315
tff(fact_10525_ATP_Olambda__316,axiom,
    ! [Uu2: nat] : aa_nat_nat(aTP_Lamm_dg,Uu2) = one_one_nat ).

% ATP.lambda_316
tff(fact_10526_ATP_Olambda__317,axiom,
    ! [Uu2: nat] : aa_nat_int(aTP_Lamm_dh,Uu2) = one_one_int ).

% ATP.lambda_317
tff(fact_10527_ATP_Olambda__318,axiom,
    ! [Uu2: int] : aa_int_int(aTP_Lamm_di,Uu2) = one_one_int ).

% ATP.lambda_318
tff(fact_10528_ATP_Olambda__319,axiom,
    ! [Uu2: real] :
      ( aa_real_o(aTP_Lamm_fp,Uu2)
    <=> $false ) ).

% ATP.lambda_319
tff(fact_10529_ATP_Olambda__320,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_fz,Uu2)
    <=> $false ) ).

% ATP.lambda_320
tff(fact_10530_ATP_Olambda__321,axiom,
    ! [Uu2: nat] :
      ( aa_nat_o(aTP_Lamm_ga,Uu2)
    <=> $true ) ).

% ATP.lambda_321

% Helper facts (11)
tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y: $o] :
      ( ( (X9) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X9: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X9)),(Y))
      | ( (X9) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ( X9 != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X9: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ( X9 != Y )
      | aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Nat__Onat_T,axiom,
    ! [X9: nat,Y: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(fequal_nat,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y: num] :
      ( ( X9 != Y )
      | aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Num__Onum_T,axiom,
    ! [X9: num,Y: num] :
      ( ~ aa_num_o(aa_num_fun_num_o(fequal_num,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fequal_2_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ( X9 != Y )
      | aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Real__Oreal_T,axiom,
    ! [X9: real,Y: real] :
      ( ~ aa_real_o(aa_real_fun_real_o(fequal_real,X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: fun_real_o] :
      ( aa_real_o(P,fChoice_real(P))
      = ( ? [X6: real] : aa_real_o(P,X6) ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,xa),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) = aa_nat_nat(aa_nat_fun_nat_nat(vEBT_VEBT_high,za),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,deg),aa_num_nat(numeral_numeral_nat,aa_num_num(bit0,one)))) ).

%------------------------------------------------------------------------------